1. Magnetic Fields and Forces
A magnetic field ($\vec{B}$) is a region of space around a magnet or a moving electric charge where a magnetic force can be detected. Magnetic field lines indicate the direction and strength of the field, emerging from north poles and entering south poles. A moving electric charge in a magnetic field experiences a magnetic force, given by the Lorentz force equation $\vec{F} = q(\vec{v} \times \vec{B})$, where $q$ is the charge, $\vec{v}$ is its velocity, and $\vec{B}$ is the magnetic field. This force is always perpendicular to both $\vec{v}$ and $\vec{B}$.
2. Motion of Charges in Magnetic Fields
When a charged particle moves in a uniform magnetic field, the magnetic force causes it to move in a circular path if its velocity is perpendicular to the field, or a helical path if there's a component of velocity parallel to the field. The magnetic force provides the centripetal force for circular motion: $q v B = \frac{mv^2}{r}$, leading to a radius of $r = \frac{mv}{qB}$. This principle is fundamental to the operation of devices like mass spectrometers and particle accelerators.
3. Magnetic Fields from Currents: Biot-Savart and Ampere's Law
Moving charges create magnetic fields. The Biot-Savart Law quantifies the magnetic field produced by a small current element: $d\vec{B} = \frac{\mu_0}{4\pi} \frac{I d\vec{l} \times \hat{r}}{r^2}$, where $\mu_0$ is the permeability of free space. Ampere's Law provides a simpler way to calculate magnetic fields for symmetric situations, relating the line integral of the magnetic field around a closed loop to the total current enclosed: $\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{\text{enc}}$. These laws are crucial for understanding electromagnetism.
4. Magnetic Fields from Currents: Straight and Solenoid (Basic)
For a long, straight wire carrying current $I$, the magnetic field at a distance $r$ from the wire is given by $B = \frac{\mu_0 I}{2\pi r}$. Inside a long solenoid with $n$ turns per unit length carrying current $I$, the magnetic field is uniform and given by $B = \mu_0 n I$. Solenoids are essential components in electromagnets, used in relays, electric bells, and magnetic locks, demonstrating the practical application of magnetic fields generated by currents.
5. Forces between Currents and Magnetic Dipoles
Current-carrying wires exert magnetic forces on each other. Two parallel wires carrying currents in the same direction attract, while those with currents in opposite directions repel. The force per unit length between two parallel wires is $F/L = \frac{\mu_0 I_1 I_2}{2\pi d}$. A magnetic dipole, like a current loop, experiences a torque in a magnetic field ($\vec{\tau} = \vec{\mu} \times \vec{B}$), where $\vec{\mu}$ is the magnetic dipole moment. This torque tends to align the dipole with the external field.
6. Electric Motors and Domestic Circuits (Basic)
Electric motors utilize the force on a current-carrying conductor in a magnetic field to produce rotational motion. A current loop in a magnetic field experiences a torque that causes it to rotate. Commutators and brushes are used to reverse the current direction periodically, ensuring continuous rotation. In domestic circuits, wiring often involves principles of electromagnetism, particularly in devices like switches, transformers, and the magnetic fields generated by currents in appliances, impacting safety and efficiency.
7. Additional: Magnetic Potential Energy of a Dipole
A magnetic dipole placed in an external magnetic field possesses magnetic potential energy. This energy depends on the orientation of the dipole's magnetic dipole moment ($\vec{\mu}$) relative to the magnetic field ($\vec{B}$). The potential energy is given by $U = -\vec{\mu} \cdot \vec{B}$. The system tends towards a state of minimum potential energy, where the dipole moment is aligned with the magnetic field. This concept is fundamental in understanding the behavior of magnets in magnetic fields and in materials science.