Introduction

The relationship between electricity and magnetism, known for over 2000 years as separate phenomena, was first empirically connected in 1820 by Danish physicist Hans Christian Oersted. He observed that an electric current in a straight wire caused a nearby magnetic compass needle to deflect, aligning itself tangentially to imaginary circles centered around the wire. This indicated that **moving charges, or electric currents, produce a magnetic field** in the space around them.

This discovery sparked intense research, culminating in James Maxwell's unification of electricity and magnetism in the 1860s, demonstrating that light is an electromagnetic wave. This understanding led to significant technological advancements in the 20th century, including the development of radio communication and various electrical devices.

This chapter explores how magnetic fields exert forces on moving charged particles and current-carrying conductors, and how electric currents generate magnetic fields.

Conventionally, a vector quantity (like current or field) pointing **out of the plane of the paper** is represented by a **dot (¤)**, like the tip of an arrow coming towards you. A vector pointing **into the plane of the paper** is represented by a **cross (Ä)**, like the feathered tail of an arrow moving away from you.

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Magnetic Force

Just as static electric charges produce an electric field ($\vec{E}$), moving electric charges (currents) produce an additional field called the **magnetic field** ($\vec{B}$). Both $\vec{E}$ and $\vec{B}$ are vector fields defined at each point in space and obey the principle of superposition (fields from multiple sources add vectorially).

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Sources And Fields

The interaction between two electric charges can be described in two stages: the source charge creates an electric field, and this field exerts a force on the second charge. Similarly, we can describe magnetic interactions in stages: a current source creates a magnetic field, and this magnetic field exerts a force on another current or moving charge.

Electric field: $\vec{E}(\vec{r}) = \frac{1}{4\pi\epsilon_0} \frac{Q}{r^2} \hat{r}$ (due to charge $Q$). Force on charge $q$: $\vec{F} = q\vec{E}$.

Magnetic field: $\vec{B}(\vec{r})$ (due to current source). This chapter details how to calculate $\vec{B}$.

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Magnetic Field, Lorentz Force

When a point charge $q$ moves with velocity $\vec{v}$ in a region with both an electric field $\vec{E}$ and a magnetic field $\vec{B}$, the total force acting on it is the sum of the electric force and the magnetic force. This total force is called the **Lorentz force**:

$\mathbf{\vec{F} = \vec{F}_{\text{electric}} + \vec{F}_{\text{magnetic}} = q\vec{E} + q(\vec{v} \times \vec{B})}$

where $\vec{v} \times \vec{B}$ is the vector (cross) product of $\vec{v}$ and $\vec{B}$.

Features of the magnetic force $ \vec{F}_{\text{magnetic}} = q(\vec{v} \times \vec{B}) $:

• It is proportional to the charge $q$, velocity $\vec{v}$, and magnetic field $\vec{B}$.
• The direction of the magnetic force is given by the right-hand rule for the cross product $\vec{v} \times \vec{B}$. If $q$ is positive, $\vec{F}_{\text{magnetic}}$ is in the direction of $\vec{v} \times \vec{B}$. If $q$ is negative, the force is in the opposite direction.
• The magnetic force is always **perpendicular** to both the velocity $\vec{v}$ and the magnetic field $\vec{B}$.
• The magnitude of the magnetic force is $|\vec{F}_{\text{magnetic}}| = |q| |\vec{v}| |\vec{B}| \sin\theta$, where $\theta$ is the angle between $\vec{v}$ and $\vec{B}$.
• The magnetic force is **zero** if the charge is stationary ($|\vec{v}|=0$). Only a moving charge experiences a magnetic force.
• The magnetic force is also **zero** if the velocity $\vec{v}$ is parallel ($\theta=0^\circ$) or antiparallel ($\theta=180^\circ$) to the magnetic field $\vec{B}$ (since $\sin 0^\circ = \sin 180^\circ = 0$).

The magnitude of the magnetic field $\vec{B}$ (magnetic field strength) is defined based on the Lorentz force. One SI unit of $\vec{B}$ is called the **tesla (T)**. One tesla is the magnetic field strength such that a charge of 1 coulomb moving perpendicular to the field with a speed of 1 m/s experiences a magnetic force of 1 newton. $1 \text{ T} = 1 \text{ N}/(\text{C} \cdot \text{m/s}) = 1 \text{ N}/(\text{A} \cdot \text{m})$.

A smaller non-SI unit, the gauss (G), is often used: $1 \text{ G} = 10^{-4} \text{ T}$. The Earth's magnetic field is about $3.6 \times 10^{-5} \text{ T}$.

Permittivity ($\epsilon$) describes how an electric field affects a medium. Permeability ($\mu$) describes how a magnetic field affects a medium. In vacuum, they are $\epsilon_0$ and $\mu_0$. They are related to the speed of light ($c$) in vacuum: $\epsilon_0\mu_0 = 1/c^2$. While $\epsilon_0$ is related to Coulomb's constant $k$ ($k=1/(4\pi\epsilon_0)$), $\mu_0$ is fixed in SI units: $\mu_0 = 4\pi \times 10^{-7}$ T m/A.

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Magnetic Force On A Current-Carrying Conductor

A current in a conductor is due to the motion of charge carriers (e.g., electrons in a metal). If a conductor is placed in an external magnetic field, the magnetic force on each moving charge carrier will contribute to a net force on the conductor itself.

Consider a straight conductor of uniform cross-sectional area $A$ and length $l$, carrying a steady current $I$. Let $n$ be the number density of mobile charge carriers (each with charge $q$) moving with an average drift velocity $\vec{v}_d$. The total number of charge carriers in the segment is $N = nAl$.

The total magnetic force on the segment is the sum of the magnetic forces on each carrier: $\vec{F} = N (q \vec{v}_d \times \vec{B}) = (nAl) q (\vec{v}_d \times \vec{B})$.

We know that the current density $\vec{j} = nq\vec{v}_d$. The current $I = |\vec{j}|A = nq|v_d|A$. We can define a vector $\vec{l}$ with magnitude $l$ and direction along the current $I$. Then $I\vec{l} = (nq\vec{v}_d)Al = \vec{j}Al$. Thus, the force on the straight conductor is:

$\mathbf{\vec{F} = I \vec{l} \times \vec{B}}$

This formula holds for a straight conductor segment of length $l$ carrying current $I$ in a uniform external magnetic field $\vec{B}$. For a wire of arbitrary shape, the total force is the vector sum (integral) of forces on infinitesimal length elements $d\vec{l}$.

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Answer:

Motion In A Magnetic Field

When a charged particle moves in a magnetic field $\vec{B}$, the magnetic force $\vec{F}_B = q(\vec{v} \times \vec{B})$ is always perpendicular to the velocity $\vec{v}$. A force perpendicular to velocity does **no work** on the particle ($W = \int \vec{F} \cdot d\vec{l} = \int \vec{F} \cdot (\vec{v} dt) = \int (\vec{F} \cdot \vec{v}) dt$). Since $\vec{F}$ is perpendicular to $\vec{v}$, $\vec{F} \cdot \vec{v} = 0$, so $W = 0$. This means the magnetic force does not change the kinetic energy or the speed of the charged particle. It only changes the direction of its velocity.

Consider a uniform magnetic field $\vec{B}$. The motion of a charged particle depends on the orientation of its velocity relative to the field.

• If $\vec{v}$ is perpendicular to $\vec{B}$: The magnetic force is $F_B = |q|vB$ and is always perpendicular to $\vec{v}$, acting as a centripetal force. This causes the particle to move in a **circular path** in a plane perpendicular to the magnetic field lines (Fig. 4.5). The radius of this circular path is determined by equating the magnetic force to the centripetal force: $|q|vB = \frac{mv^2}{r}$. So, the radius is $\mathbf{r = \frac{mv}{|q|B}}$. The larger the particle's momentum ($mv$), the larger the radius.
• If $\vec{v}$ has a component parallel to $\vec{B}$ ($v_{||}$) and a component perpendicular to $\vec{B}$ ($v_{\perp}$): The magnetic force acts only on the perpendicular component of velocity ($F_B = |q|v_{\perp}B$). This force causes circular motion in the plane perpendicular to $\vec{B}$ (with radius $r = \frac{mv_{\perp}}{|q|B}$). The parallel component of velocity ($v_{||}$) is unaffected by the magnetic field, so the particle moves along the field line with constant velocity $v_{||}$. The combination of uniform circular motion and uniform linear motion along the axis creates a **helical path** (Fig. 4.6).

For the circular motion component (whether purely circular or part of a helix), the angular frequency $\omega$ and frequency $\nu$ are related to the period $T = 2\pi r/v_{\perp}$ by $\omega = v_{\perp}/r$ and $\nu = 1/T = \omega/(2\pi)$. Using $r = \frac{mv_{\perp}}{|q|B}$, we get $v_{\perp}/r = \frac{|q|B}{m}$.

So, $\mathbf{\omega = \frac{|q|B}{m}}$ and $\mathbf{\nu = \frac{|q|B}{2\pi m}}$.

This frequency (or angular frequency) is called the **cyclotron frequency** and is remarkable because it is **independent of the particle's speed ($v_{\perp}$) and the radius ($r$)** of the circular orbit. This property is crucial in particle accelerators like the cyclotron.

The distance moved parallel to the field during one complete revolution of the circular motion is called the **pitch** ($p$) of the helix: $p = v_{||} T = v_{||} (2\pi/\omega) = v_{||} \frac{2\pi m}{|q|B}$.

Answer:

Magnetic Field Due To A Current Element, Biot-Savart Law

The Biot-Savart law is a fundamental law that allows us to calculate the magnetic field produced by a small segment of a current-carrying conductor. It is the magnetic analogue of Coulomb's law for electric fields.

Consider a small element of a conductor of length $d\vec{l}$ carrying a steady current $I$. This current element is considered the source of the magnetic field. Let $\vec{r}$ be the displacement vector from the current element $d\vec{l}$ to a point P where we want to determine the magnetic field $d\vec{B}$. Let $\theta$ be the angle between $d\vec{l}$ and $\vec{r}$.

According to the Biot-Savart law, the magnitude of the magnetic field $d\vec{B}$ produced by this current element at point P is directly proportional to the current $I$, the length $|d\vec{l}|$, and $\sin\theta$, and inversely proportional to the square of the distance $r = |\vec{r}|$. The direction of $d\vec{B}$ is perpendicular to the plane containing $d\vec{l}$ and $\vec{r}$, and is given by the right-hand rule for the cross product $d\vec{l} \times \vec{r}$.

In vector notation, the Biot-Savart law is:

$\mathbf{d\vec{B} = \frac{\mu_0}{4\pi} \frac{I d\vec{l} \times \vec{r}}{r^3}}$

where $\mu_0$ is the permeability of free space (vacuum). The constant $\frac{\mu_0}{4\pi}$ has an exact value in SI units: $\frac{\mu_0}{4\pi} = 10^{-7}$ T m/A.

The magnitude of $d\vec{B}$ is:

$\mathbf{|d\vec{B}| = \frac{\mu_0}{4\pi} \frac{I dl \sin\theta}{r^2}}$

The total magnetic field $\vec{B}$ at point P due to an entire current-carrying conductor is obtained by integrating $d\vec{B}$ over the entire length of the conductor:

$\mathbf{\vec{B} = \int d\vec{B} = \int \frac{\mu_0}{4\pi} \frac{I d\vec{l} \times \vec{r}}{r^3}}$

Similarities and differences between Biot-Savart law and Coulomb's law:

• Both are inverse-square laws (magnitude proportional to $1/r^2$). Both obey the superposition principle. Both are linear in their sources.
• Coulomb's law source is a scalar (charge $q$). Biot-Savart law source is a vector ($I d\vec{l}$).
• Electric field is along the displacement vector $\vec{r}$. Magnetic field is perpendicular to the plane containing $d\vec{l}$ and $\vec{r}$.
• Biot-Savart law has a $\sin\theta$ dependence (or cross product dependence); the field is zero along the direction of $d\vec{l}$. Coulomb's law has no such angular dependence on the displacement direction relative to the source (except implicit in $r$).

Answer:

Magnetic Field On The Axis Of A Circular Current Loop

We can use the Biot-Savart law to calculate the magnetic field produced by a circular loop carrying a steady current $I$. Consider a loop of radius $R$ in the y-z plane with its center at the origin. We want to find the field at a point P on the x-axis at a distance $x$ from the center.

Take a small current element $d\vec{l}$ on the loop. The displacement vector $\vec{r}$ from $d\vec{l}$ to P has magnitude $r = \sqrt{x^2 + R^2}$. The vector $d\vec{l}$ is always perpendicular to the vector from the center of the loop to the element (radius vector of the loop). For an axial point P, $d\vec{l}$ is also perpendicular to $\vec{r}$, so $\theta = 90^\circ$ and $|d\vec{l} \times \vec{r}| = dl \cdot r$.

The magnetic field $d\vec{B}$ due to $d\vec{l}$ is perpendicular to the plane containing $d\vec{l}$ and $\vec{r}$. For an element on the loop, $d\vec{B}$ has a component along the x-axis ($dB_x$) and a component perpendicular to the x-axis ($dB_{\perp}$). Due to symmetry, when we integrate over the entire loop, the perpendicular components $dB_{\perp}$ cancel out. Only the axial components $dB_x$ add up.

$dB_x = |d\vec{B}| \cos\phi$, where $\phi$ is the angle between $d\vec{B}$ and the x-axis. From the geometry, $\cos\phi = R/r = R/\sqrt{x^2+R^2}$.

The magnitude of $d\vec{B}$ is $|d\vec{B}| = \frac{\mu_0}{4\pi} \frac{I dl \sin 90^\circ}{r^2} = \frac{\mu_0}{4\pi} \frac{I dl}{x^2+R^2}$.

$dB_x = \frac{\mu_0}{4\pi} \frac{I dl}{x^2+R^2} \frac{R}{\sqrt{x^2+R^2}} = \frac{\mu_0}{4\pi} \frac{IR dl}{(x^2+R^2)^{3/2}}$.

To find the total field $\vec{B}$ at P, integrate $dB_x$ over the entire loop. The terms $I, R, x, \mu_0$ are constant for all elements. The integral of $dl$ over the loop is the circumference $2\pi R$.

$B_x = \int dB_x = \int \frac{\mu_0}{4\pi} \frac{IR dl}{(x^2+R^2)^{3/2}} = \frac{\mu_0 IR}{(x^2+R^2)^{3/2}} \int dl = \frac{\mu_0 IR (2\pi R)}{(x^2+R^2)^{3/2}} = \frac{\mu_0 I (2\pi R^2)}{4\pi (x^2+R^2)^{3/2}}$.

The magnetic field on the axis of a circular current loop is directed along the axis (positive x-direction if current is anticlockwise when viewed from positive x-axis, given by the right-hand rule) and its magnitude is:

$\mathbf{B_x = \frac{\mu_0 I R^2}{2 (x^2+R^2)^{3/2}}}$

At the center of the loop, $x=0$:

$\mathbf{B_{\text{center}} = \frac{\mu_0 I R^2}{2 (0^2+R^2)^{3/2}} = \frac{\mu_0 I R^2}{2 R^3} = \frac{\mu_0 I}{2R}}$

The direction of the magnetic field produced by a current loop can be found using the **right-hand thumb rule for circular currents**: Curl the fingers of your right hand in the direction of the current in the loop; your extended thumb points in the direction of the magnetic field through the loop.

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Answer:

Ampere’S Circuital Law

Ampere's circuital law provides an alternative way to relate the magnetic field $\vec{B}$ to the electric current $I$ that produces it. It is analogous to Gauss's law in electrostatics, which relates the electric field to the enclosed charge.

Ampere's law states that the **line integral of the magnetic field ($\vec{B}$) around any closed loop (called an Amperian loop)** is proportional to the **total steady current ($I_{enc}$) passing through the surface bounded by the loop**. The constant of proportionality is the permeability of free space ($\mu_0$).

$\mathbf{\oint_C \vec{B} \cdot d\vec{l} = \mu_0 I_{enc}}$

Here, the integral is taken around the closed loop $C$, $d\vec{l}$ is an infinitesimal element of length along the loop, and $I_{enc}$ is the net current passing through any open surface $S$ that has the loop $C$ as its boundary. The direction of $I_{enc}$ is related to the direction of traversal of the loop by the right-hand rule: if you curl the fingers of your right hand in the direction the loop is traversed, your thumb points in the direction of positive current through the surface.

In situations with high symmetry, we can choose an Amperian loop such that the magnetic field $\vec{B}$ has a constant magnitude tangential to the loop along its length $L$. In such cases, $\oint \vec{B} \cdot d\vec{l} = B \oint dl = BL$. The law then simplifies to:

$\mathbf{BL = \mu_0 I_{enc}}$

This simplified form is very useful for calculating the magnetic field when the system has sufficient symmetry, similar to how Gauss's law simplifies electric field calculations for symmetric charge distributions.

Example: For a long straight wire carrying current $I$, we can choose a circular Amperian loop of radius $r$ centered on the wire and in a plane perpendicular to the wire. By symmetry, $\vec{B}$ is tangential to the loop and has constant magnitude $B$ around the loop. The length of the loop is $L = 2\pi r$. The enclosed current is $I_{enc} = I$. Applying the simplified Ampere's law: $B (2\pi r) = \mu_0 I$. So, $\mathbf{B = \frac{\mu_0 I}{2\pi r}}$. This is the magnitude of the magnetic field at a distance $r$ from a long straight wire. The field lines are concentric circles around the wire.

The **right-hand rule for a straight wire** gives the direction of $\vec{B}$: Grasp the wire with your right hand, with your thumb pointing in the direction of the current. Your fingers will curl around the wire in the direction of the magnetic field lines.

Ampere's circuital law and the Biot-Savart law are equivalent for steady currents. Ampere's law is often more convenient when the current distribution has high symmetry. Ampere's law holds for steady currents (not time-varying).

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The Solenoid And The Toroid

Solenoids and toroids are current configurations used to generate strong, uniform magnetic fields in specific regions. Ampere's law is particularly useful for calculating the field produced by ideal versions of these devices due to their symmetry.

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The Solenoid

A **solenoid** is a long coil consisting of a large number of closely packed turns of wire wound in the form of a helix. A long solenoid is one whose length is much greater than its radius.

When a current flows through a solenoid, the magnetic fields due to the individual turns add up. For an ideal long solenoid, the magnetic field lines are nearly uniform, strong, and parallel to the axis inside the solenoid, and they are very weak outside. The field outside an ideal infinite solenoid is zero.

To calculate the magnetic field $\vec{B}$ inside a long solenoid, we use Ampere's circuital law and choose a rectangular Amperian loop with one side inside the solenoid and parallel to the axis, and the other sides perpendicular to the axis or outside the solenoid (Fig. 4.18). Let the side inside the solenoid have length $h$ and be parallel to the axis where the field is uniform ($B$).

Applying $\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc}$ around the loop:

• The integral along the side inside and parallel to the axis is $B \times h$ (since $\vec{B}$ and $d\vec{l}$ are parallel).
• The integral along the sides perpendicular to the axis is zero (since $\vec{B}$ is parallel to the axis, it is perpendicular to these sides, so $\vec{B} \cdot d\vec{l} = 0$).
• The integral along the side outside the solenoid is zero (since the field outside an ideal long solenoid is zero).

So, the total line integral is $B h$.

The current enclosed by the loop is the total current passing through the surface bounded by the loop. If $n$ is the number of turns per unit length of the solenoid, the number of turns in length $h$ is $nh$. If the current in each turn is $I$, the total current enclosed is $I_{enc} = (nh) I$.

Applying Ampere's law: $B h = \mu_0 (nhI)$.

Solving for $B$: $\mathbf{B = \mu_0 n I}$.

The magnitude of the magnetic field inside a long solenoid is uniform and depends only on the number of turns per unit length and the current. The direction is along the axis, given by the right-hand rule for solenoids: Curl your fingers in the direction of the current in the windings; your thumb points in the direction of the magnetic field inside the solenoid.

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The Toroid

A **toroid** is a hollow circular ring (doughnut shape) on which a large number of turns of wire are closely wound. It can be seen as a long solenoid bent into a closed circular shape.

To calculate the magnetic field, consider circular Amperian loops centered on the axis of the toroid. Due to the toroid's symmetry, the magnetic field is tangential to these loops and has a constant magnitude on any given loop.

• For a loop in the empty space inside the toroid (point P, loop 1 in Fig. 4.19(b)): The loop encloses no current. By Ampere's law, $\oint \vec{B} \cdot d\vec{l} = B_1 (2\pi r_1) = \mu_0 (0) = 0$. So, the magnetic field inside the empty space of the toroid is zero ($B_1=0$).
• For a loop in the empty space outside the toroid (point Q, loop 3 in Fig. 4.19(b)): This loop encloses all the turns. However, for each turn, the current passes through the surface bounded by the loop twice, once coming out of the paper and once going into the paper, with opposite directions relative to the surface normal (determined by the loop direction). The net current enclosed by the loop is zero ($I_{enc} = 0$). By Ampere's law, $\oint \vec{B} \cdot d\vec{l} = B_3 (2\pi r_3) = \mu_0 (0) = 0$. So, the magnetic field outside the toroid is also zero ($B_3=0$).
• For a loop inside the core of the toroid (point S, loop 2 in Fig. 4.19(b)) at an average radius $r$: The magnetic field $\vec{B}$ is tangential and has a constant magnitude $B(r)$. The length of the loop is $2\pi r$. The total current enclosed is the current $I$ flowing through each of the $N$ turns. So, $I_{enc} = NI$.

Applying Ampere's law: $\oint \vec{B} \cdot d\vec{l} = B(r) (2\pi r) = \mu_0 (NI)$.

Solving for $B(r)$: $\mathbf{B(r) = \frac{\mu_0 NI}{2\pi r}}$.

For an ideal toroid with closely wound turns and uniform cross-section, $r$ can be taken as the average radius of the toroid. If $n$ is the number of turns per unit length ($n = N/(2\pi r)$), then $N = n(2\pi r)$, and the formula becomes $B = \mu_0 n I$, which is the same as for a long solenoid. Thus, a toroid provides a way to create a uniform magnetic field confined entirely within the doughnut shape.

Answer:

Force Between Two Parallel Currents, The Ampere

We've established that a current creates a magnetic field (Biot-Savart law, Ampere's law) and that a magnetic field exerts a force on a current-carrying conductor (Lorentz force on charge carriers). Therefore, two current-carrying conductors near each other will exert magnetic forces on each other.

Consider two long, straight parallel conductors 'a' and 'b', separated by a distance $d$, carrying steady currents $I_a$ and $I_b$ respectively. Assume the currents are in the same direction (parallel).

Conductor 'a' creates a magnetic field $\vec{B}_a$ at the location of conductor 'b'. Using the right-hand rule for a straight wire, the direction of $\vec{B}_a$ at 'b' is into the plane of the paper (if wires are horizontal and current is to the right). The magnitude is $B_a = \frac{\mu_0 I_a}{2\pi d}$.

Conductor 'b' carries current $I_b$ in this field $\vec{B}_a$. The force on a segment of length $L$ of conductor 'b' is $\vec{F}_{ba} = I_b \vec{L} \times \vec{B}_a$. The vector $\vec{L}$ is in the direction of $I_b$, and $\vec{B}_a$ is perpendicular to $\vec{L}$. The direction of $\vec{F}_{ba}$ is given by $\vec{L} \times \vec{B}_a$, which is towards conductor 'a' (if current is right, B is into paper, force is up/towards 'a'). So, parallel currents attract each other.

The magnitude of the force $\vec{F}_{ba}$ is $F_{ba} = I_b L B_a \sin 90^\circ = I_b L \frac{\mu_0 I_a}{2\pi d}$.

$\mathbf{F_{ba} = \frac{\mu_0 I_a I_b}{2\pi d} L}$

Similarly, conductor 'b' creates a field $\vec{B}_b$ at 'a', which exerts a force $\vec{F}_{ab}$ on 'a'. $F_{ab} = F_{ba}$, and $\vec{F}_{ab}$ is directed towards 'b'. So $\vec{F}_{ba} = -\vec{F}_{ab}$, consistent with Newton's third law.

If the currents are in opposite directions (antiparallel), one can show that the conductors repel each other.

Summary: **Parallel currents attract, and antiparallel currents repel.**

This is the opposite of electrostatics where like charges repel and unlike charges attract.

The force per unit length ($f$) between the two conductors is $f = F/L$.

$\mathbf{f = \frac{\mu_0 I_a I_b}{2\pi d}}$

This expression is used to define the **ampere (A)**, the SI base unit of electric current:

**One ampere is the constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed one metre apart in vacuum, would produce between these conductors a force equal to $2 \times 10^{-7}$ newtons per metre of length.**

This definition was adopted in 1946. The SI unit of charge, the coulomb (C), is then defined as $1 \text{ C} = 1 \text{ A} \cdot \text{s}$.

Answer:

Torque On Current Loop, Magnetic Dipole

A current loop placed in an external magnetic field experiences forces on its segments, which can result in a net torque on the loop. A current loop also behaves like a magnetic dipole and produces a magnetic field similar to that of an electric dipole at large distances.

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Torque On A Rectangular Current Loop In A Uniform Magnetic Field

Consider a rectangular loop of sides $a$ and $b$, carrying current $I$, placed in a uniform magnetic field $\vec{B}$. Let the sides of length $b$ be parallel to the y-axis and the sides of length $a$ parallel to the x-axis. The magnetic field $\vec{B}$ is uniform.

Forces on the sides parallel to y-axis: The length vector $\vec{l}$ is along $\pm \hat{j}$. If $\vec{B}$ is in the x-z plane, $\vec{l} \times \vec{B}$ is in the x-z plane and perpendicular to $\vec{l}$. The force on AD is $\vec{F}_{AD} = I(b\hat{j}) \times \vec{B}$. The force on BC is $\vec{F}_{BC} = I(-b\hat{j}) \times \vec{B}$. These forces are equal and opposite and are collinear along the axis connecting the centers of AD and BC. They cancel each other, producing no net force or torque.

Forces on the sides parallel to x-axis (AB and CD): The length vector $\vec{l}$ is along $\pm \hat{i}$. Let the magnetic field $\vec{B}$ be uniform. Consider the case where $\vec{B}$ is in the plane of the loop, along the x-axis (Fig. 4.21(a)). Force on AB: $\vec{F}_1 = I(a\hat{i}) \times (B\hat{i}) = 0$. Force on CD: $\vec{F}_2 = I(-a\hat{i}) \times (B\hat{i}) = 0$. This is incorrect, if B is in the plane of the loop, it must be perpendicular to the current segments for force. Let the magnetic field $\vec{B}$ be along the x-axis (length $a$ is along x). Then the sides of length $b$ are parallel to the y-axis. Side AB has current $I$ along $\hat{i}$, side CD has current $I$ along $-\hat{i}$. Sides AD and BC have current along $\pm \hat{j}$. If $\vec{B}$ is along $\hat{i}$, then force on AB and CD is zero ($I \vec{l} \times \vec{B} = I (a\hat{i}) \times (B\hat{i}) = 0$). Force on AD: $\vec{F}_{AD} = I(b\hat{j}) \times (B\hat{i}) = -IbB\hat{k}$ (into page). Force on BC: $\vec{F}_{BC} = I(-b\hat{j}) \times (B\hat{i}) = IbB\hat{k}$ (out of page). These forces are equal and opposite and form a couple. The perpendicular distance between them is $a$. Torque magnitude $\tau = IbB \times a = IabB = IAB$, where $A=ab$ is the area. This torque rotates the loop around the x-axis.

Consider the general case where the normal to the plane of the loop (area vector $\vec{A}$) makes an angle $\theta$ with the uniform magnetic field $\vec{B}$. The forces on the sides parallel to $\vec{B}$ are zero. The forces on the sides perpendicular to $\vec{B}$ form a couple. The magnitude of the force on each side is $F = IbB$. The perpendicular distance between these forces is $a \sin\theta$.

The magnitude of the torque is $\tau = (IbB)(a \sin\theta) = I(ab)B\sin\theta = IAB\sin\theta$.

We define the **magnetic moment** $\vec{m}$ of the current loop as a vector whose magnitude is $IA$ (current times area) and whose direction is perpendicular to the plane of the loop, given by the right-hand thumb rule (curl fingers in current direction, thumb points along $\vec{m}$).

$\mathbf{\vec{m} = I \vec{A}}$

where $\vec{A}$ is the area vector. The magnitude is $m=IA$.

In terms of the magnetic moment, the torque on the loop is:

$\mathbf{\vec{\tau} = \vec{m} \times \vec{B}}$

The magnitude is $\tau = mB\sin\theta$, where $\theta$ is the angle between $\vec{m}$ and $\vec{B}$.

The torque is zero when $\vec{m}$ is parallel ($\theta=0^\circ$) or antiparallel ($\theta=180^\circ$) to $\vec{B}$. $\theta=0^\circ$ is a **stable equilibrium** (any small rotation creates a restoring torque). $\theta=180^\circ$ is an **unstable equilibrium**. This explains why a magnetic dipole (like a compass needle or bar magnet) aligns itself with an external magnetic field.

If the loop has $N$ closely wound turns, the total magnetic moment is $N$ times the moment of a single turn: $\mathbf{\vec{m} = N I \vec{A}}$, and the torque is $\mathbf{\vec{\tau} = N I \vec{A} \times \vec{B} = \vec{m} \times \vec{B}}$.

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Circular Current Loop As A Magnetic Dipole

We have calculated the magnetic field on the axis of a circular current loop. For distances $x \gg R$, the formula $B_x = \frac{\mu_0 I R^2}{2 (x^2+R^2)^{3/2}}$ simplifies to $B_x \approx \frac{\mu_0 I R^2}{2 x^3}$. Using the area $A=\pi R^2$, $R^2 = A/\pi$, so $B_x \approx \frac{\mu_0 I (A/\pi)}{2 x^3} = \frac{\mu_0 I A}{2\pi x^3}$. Defining the magnitude of magnetic moment $m = IA$, we get $\mathbf{B_x \approx \frac{\mu_0 m}{2\pi x^3}}$ for $x \gg R$.

For a point magnetic dipole $\vec{m}$, the magnetic field at a point on its axis is exactly $\vec{B}_{\text{axis}} = \frac{\mu_0}{2\pi} \frac{\vec{m}}{x^3}$. This confirms that a circular current loop acts as a magnetic dipole at large distances.

The magnetic field on the equatorial plane of a point magnetic dipole at distance $x$ is exactly $\vec{B}_{\text{equatorial}} = \frac{\mu_0}{4\pi} \frac{-\vec{m}}{x^3}$. This is analogous to the electric field of an electric dipole on its equatorial plane $\vec{E}_{\text{equatorial}} = \frac{1}{4\pi\epsilon_0} \frac{-\vec{p}}{x^3}$. The analogy is strong: $\mu_0 \leftrightarrow 1/\epsilon_0$ and $\vec{m} \leftrightarrow \vec{p}$.

A key difference: electric dipoles are made of separate positive and negative charges (electric monopoles). Magnetic monopoles are not known to exist. The most elementary magnetic structure observed is a magnetic dipole (or a current loop).

Ampere hypothesised that all magnetism is due to circulating currents, which explains many phenomena, but elementary particles like electrons and protons also have an intrinsic magnetic moment (spin magnetic moment) not fully explained by classical circulating currents.

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The Magnetic Dipole Moment Of A Revolving Electron

In the Bohr model of the hydrogen atom, an electron (charge $-e$) revolves in a circular orbit of radius $r$ around the nucleus. This constitutes a current $I$. If the electron speed is $v$ and period is $T = 2\pi r/v$, the current is $I = e/T = ev/(2\pi r)$.

This circulating current creates an orbital magnetic dipole moment $\vec{\mu}_l$. Its magnitude is $\mu_l = IA = I(\pi r^2) = \frac{ev}{2\pi r}(\pi r^2) = \frac{1}{2} evr$. The direction is perpendicular to the orbit plane. By the right-hand rule (current direction is opposite to electron velocity), if the electron revolves anticlockwise, the current is clockwise, and $\vec{\mu}_l$ points into the plane. The angular momentum vector $\vec{l}$ for anticlockwise electron motion is out of the plane. So $\vec{\mu}_l$ is opposite to $\vec{l}$.

$\mathbf{\vec{\mu}_l = -\frac{e}{2m_e} \vec{l}}$

where $m_e$ is the electron mass and $\vec{l} = m_e \vec{v} \times \vec{r}$ is the orbital angular momentum. The ratio $\frac{\mu_l}{l} = \frac{e}{2m_e}$ is called the **gyromagnetic ratio** for orbital motion. Its value is about $8.8 \times 10^{10}$ C/kg.

According to Bohr's quantisation condition, the angular momentum $l$ can only take discrete values $l = n \frac{h}{2\pi}$ (where $n=1, 2, 3, ...$ and $h$ is Planck's constant). For $n=1$, the minimum orbital angular momentum is $l_{min} = \frac{h}{2\pi}$. The corresponding minimum orbital magnetic moment is $\mu_{l\_min} = \frac{e}{2m_e} l_{min} = \frac{e}{2m_e} \frac{h}{2\pi} = \frac{eh}{4\pi m_e}$.

This value is called the **Bohr magneton** ($\mu_B$), representing the elementary unit of atomic magnetic moment:

$\mathbf{\mu_B = \frac{eh}{4\pi m_e} \approx 9.27 \times 10^{-24} \text{ A m}^2 \text{ or J/T}}$.

Besides this orbital magnetic moment, electrons also possess an intrinsic **spin magnetic moment**, which has a value equal to the Bohr magneton. This intrinsic moment is not associated with classical spinning but is a fundamental property of the electron. This contributes to the magnetic properties of materials.

The Moving Coil Galvanometer

The **moving coil galvanometer (MCG)** is an instrument used to detect or measure small electric currents and voltages. Its operation is based on the principle that a current-carrying coil placed in a magnetic field experiences a torque.

An MCG consists of a coil with many turns, typically wound on a non-magnetic frame, suspended or pivoted in a uniform radial magnetic field produced by a permanent magnet. A cylindrical soft iron core is placed inside the coil to make the field radial and increase its strength.

When a current $I$ flows through the coil, it experiences a magnetic torque $\vec{\tau} = \vec{m} \times \vec{B}$, where $\vec{m} = NI\vec{A}$ is the magnetic moment of the coil. In a radial field, $\vec{B}$ is always perpendicular to the plane of the coil ($\vec{A}$) regardless of the coil's rotation angle (relative to the field lines). Thus, $\sin\theta = 1$, and the magnitude of the magnetic torque is $\tau_{magnetic} = NIAB$, where $A$ is the area of the coil.

This magnetic torque causes the coil to rotate. The rotation is opposed by a counter-torque provided by a spring (or suspension wire) connected to the coil. The restoring torque of the spring is proportional to the angular deflection $\phi$: $\tau_{restoring} = k\phi$, where $k$ is the torsional constant of the spring.

In equilibrium, the magnetic torque is balanced by the restoring torque:

$NIAB = k\phi$

The deflection $\phi$ is directly proportional to the current $I$:

$\mathbf{\phi = \left(\frac{NAB}{k}\right) I}$

The deflection $\phi$ is indicated by a pointer attached to the coil or suspension. The scale is calibrated to measure current or voltage.

The quantity $\frac{NAB}{k}$ is a constant for a given galvanometer. The **current sensitivity** is defined as the deflection per unit current: $\frac{\phi}{I} = \frac{NAB}{k}$. Increasing $N$, $A$, or $B$, or decreasing $k$ increases current sensitivity.

A galvanometer is a sensitive current detector. To convert it into an **ammeter** (to measure larger currents), a small resistance called a **shunt resistance ($r_s$)** is connected **in parallel** with the galvanometer coil. This diverts most of the current through the shunt, while only a small fraction passes through the galvanometer. The ammeter must have very low total resistance to be connected in series in a circuit without significantly changing the current it measures.

To convert a galvanometer into a **voltmeter** (to measure voltage), a large resistance $R$ is connected ** in series** with the galvanometer coil. This limits the current flowing through the galvanometer when it is connected in parallel across a circuit element to measure voltage. A voltmeter must have very high total resistance to draw negligible current from the circuit it is measuring, so as not to alter the voltage drop it is intended to measure.

The **voltage sensitivity** is the deflection per unit voltage: $\frac{\phi}{V} = \frac{\phi}{IR_G} = \frac{NAB}{kR_G}$, where $R_G$ is the resistance of the galvanometer coil. Note that increasing current sensitivity by increasing $N$ (which also increases $R_G$) might not increase voltage sensitivity.

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Summary

This chapter explores the magnetic effects of moving charges and currents.

• **Lorentz force** on a charge $q$ moving with velocity $\vec{v}$ in fields $\vec{E}$ and $\vec{B}$ is $\vec{F} = q(\vec{E} + \vec{v} \times \vec{B})$. Magnetic force is $q(\vec{v} \times \vec{B})$, perpendicular to $\vec{v}$ and $\vec{B}$. It does no work, only changes direction.
• Magnetic force on straight conductor of length $\vec{l}$ carrying current $I$ in field $\vec{B}$ is $\vec{F} = I \vec{l} \times \vec{B}$.
• Charged particle motion in uniform $\vec{B}$ is circular if $\vec{v} \perp \vec{B}$ (radius $r = mv/|q|B$) or helical if $\vec{v}$ has a parallel component. Cyclotron frequency $\nu = |q|B/(2\pi m)$ is speed/radius independent.
• Combined $\vec{E}$ and $\vec{B}$ fields can act as a **velocity selector** ($v = E/B$).
• **Cyclotron** accelerates charged particles to high energies using crossed $\vec{E}$ and $\vec{B}$ fields and resonance.
• **Biot-Savart law** gives magnetic field $d\vec{B}$ from current element $I d\vec{l}$: $d\vec{B} = \frac{\mu_0}{4\pi} \frac{I d\vec{l} \times \vec{r}}{r^3}$. Total field is integral.
• Field on axis of circular loop (radius $R$, current $I$) at distance $x$: $B_x = \frac{\mu_0 I R^2}{2(x^2+R^2)^{3/2}}$. At center: $B = \frac{\mu_0 I}{2R}$. Direction by right-hand rule.
• **Ampere's Circuital Law:** $\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc}$ (for steady currents). Useful for symmetric cases: $BL = \mu_0 I_{enc}$.
• Field from long straight wire (current $I$, distance $r$): $B = \frac{\mu_0 I}{2\pi r}$. Field lines are concentric circles. Direction by right-hand thumb rule.
• Field inside long solenoid (n turns/length, current $I$): $B = \mu_0 n I$. Field inside toroid (N turns, average radius $r$, current $I$): $B = \frac{\mu_0 NI}{2\pi r} = \mu_0 n I$. Field is uniform inside, zero outside.
• **Force between parallel currents:** Parallel currents attract ($f = \frac{\mu_0 I_a I_b}{2\pi d}$), antiparallel currents repel. Used to define the Ampere.
• **Torque on current loop:** Rectangular loop in uniform $\vec{B}$ experiences torque $\vec{\tau} = \vec{m} \times \vec{B}$, where $\vec{m} = NI\vec{A}$ is magnetic moment. Net force is zero. Stable equilibrium when $\vec{m} || \vec{B}$.
• Circular current loop acts as a **magnetic dipole** at large distances. Analogy with electric dipole. Magnetic monopoles not observed.
• Magnetic dipole moment of revolving electron: $\vec{\mu}_l = -\frac{e}{2m_e} \vec{l}$. Gyromagnetic ratio $\frac{e}{2m_e}$. Minimum moment is Bohr magneton $\mu_B$.
• **Moving Coil Galvanometer (MCG):** Detects/measures current using magnetic torque on a coil in a magnetic field. Deflection $\phi \propto I$. Can be converted to ammeter (shunt $r_s$ in parallel) or voltmeter (large $R$ in series).

Exercises

Questions covering calculations of magnetic force, magnetic fields due to various current configurations (wires, loops, solenoids), particle motion in magnetic fields, applications like velocity selectors and cyclotrons, torque on current loops, properties of magnetic dipoles, and principles of the moving coil galvanometer and its conversion to ammeter/voltmeter.

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ADDITIONAL EXERCISES

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