Electromagnetism: Magnetic Fields, Circuits, and Current

Magnetic Fields and Units

SI Units of Magnetic Field

The magnetic field, denoted as B, is a fundamental quantity in electromagnetism. Its SI unit is the tesla (T), which can be expressed in basic SI units as:

• Unit of B: $\text{kg} \cdot \text{s}^{-2} \cdot \text{A}^{-1}$

Magnetic Dipole Moment

The magnetic dipole moment is a vector quantity that characterizes the strength and orientation of a magnet or current loop.

• Unit: $\text{A} \cdot \text{m}^2$

Superconductivity and Magnetic Materials

Superconductors

Superconductors are materials that, below a certain critical temperature, undergo a phase transition and lose all electrical resistance. They also expel magnetic fields (Meissner effect).

• Superconducting state: Zero resistance, expulsion of magnetic fields.

Ferromagnetic and Diamagnetic Materials

• Copper (Cu): Diamagnetic (repels magnetic fields).

• Nickel (Ni): Ferromagnetic (strongly attracted to magnetic fields).

Ferromagnetic materials have a high relative permeability $\mu_r$ (typically between 0.99 and 1.01).

Kirchhoff's Laws and Electric Circuits

Kirchhoff's Loop Rule

Kirchhoff's loop rule states that the sum of all potential differences around a closed loop is zero:

• $\sum \Delta V = 0$

This is a consequence of energy conservation in electrical circuits.

Current in a Closed Loop

The current $I$ through a closed loop is related to the circulation of the magnetic field around the loop:

• $\oint \vec{B} \cdot d\vec{l}$

This is a statement of Ampère's Law.

Conservative Fields and Potential

Conservative Nature of Electric Fields

The electric field is conservative if:

• $\vec{\nabla} \times \vec{E} = 0$

This applies to static electric fields (electrostatics).

Electric Potential

The electric potential $V$ is related to the electric field $\vec{E}$ by:

• $\vec{E} = -\vec{\nabla} V$

Capacitors and RC Circuits

Capacitor Charging and Discharging

When a capacitor is charged or discharged in an RC circuit, the charge $Q$ and current $I$ follow exponential laws.

• Equation for RC circuit: $Q = Q_0 e^{-t/RC}$

• Time constant: $\tau = RC$

Current and Charge Relations

• Current: $I = \frac{dQ}{dt}$

Magnetic Field Due to Currents

Magnetic Field Inside a Wire

The magnetic field inside a long straight wire carrying current $I$ is given by:

• $B = \frac{\mu_0 I r}{2 \pi a^2}$ (for $r < a$)

Where $a$ is the radius of the wire, $r$ is the distance from the center, and $\mu_0$ is the permeability of free space.

Magnetic Field Outside a Wire

• $B = \frac{\mu_0 I}{2 \pi r}$ (for $r > a$)

Force Between Parallel Currents

Parallel wires carrying currents exert forces on each other:

• Force per unit length: $F/L = \frac{\mu_0 I_1 I_2}{2 \pi d}$

Like currents attract; opposite currents repel.

Current Density

Definition and Direction

Current density $\vec{J}$ is a vector quantity representing the flow of electric charge per unit area:

• $\vec{J} = I / (\pi a^2) \hat{z}$

Current density is a vector; current is a scalar.

Summary Table: Magnetic Field Formulas

Situation	Formula	Notes
Inside wire ($r < a$)	$B = \frac{\mu_0 I r}{2 \pi a^2}$	Linear with $r$
Outside wire ($r > a$)	$B = \frac{\mu_0 I}{2 \pi r}$	Decreases with $r$
Between parallel wires	$F/L = \frac{\mu_0 I_1 I_2}{2 \pi d}$	Attractive if currents are same direction

Additional info:

• Some questions refer to topics not yet covered (as noted in the original file).

• Explanations have been expanded for clarity and completeness.