Magnetic Fields, Forces, and Electromagnetic Induction: Study Notes

Magnetic Fields and Their Sources

Magnetic Field from Current-Carrying Wires

The magnetic field generated by a current-carrying wire is fundamental to electromagnetism. The direction of the field can be determined using the right-hand rule, and the field forms concentric circles around the wire.

• Right-Hand Rule: Point your thumb in the direction of the current; your fingers curl in the direction of the magnetic field lines.

• Compass Response: A compass placed near a current-carrying wire aligns tangentially to the magnetic field lines.

• Equation for a Long Straight Wire: $B_{\text{wire}} = \frac{\mu_0}{2\pi} \frac{I}{r}$

• Superposition Principle: The net magnetic field from multiple sources is the vector sum of individual fields. $B_{\text{total}} = B_1 + B_2 + \cdots$

Magnetic Field from Moving Charges

A moving charge produces a magnetic field, whose magnitude and direction depend on the charge's velocity and position relative to the observation point.

• Biot-Savart Law for Point Charge: $B_{\text{point charge}} = \frac{\mu_0}{4\pi} \frac{q v \sin \theta}{r^2}$

• Vector Form: $\vec{B}_{\text{point charge}} = \frac{\mu_0}{4\pi} \frac{q \vec{v} \times \hat{r}}{r^2}$

• Direction: Determined by the right-hand rule for cross products.

Magnetic Fields from Current Loops and Dipoles

Current Loops and Magnetic Dipoles

Current loops generate magnetic fields similar to those of permanent magnets. The magnetic dipole moment is a vector quantity associated with the loop.

• Magnetic Dipole Moment: $m = A I$ (where $A$ is area, $I$ is current)

• Field on Axis: $B_{\text{dipole}} = \frac{\mu_0}{4\pi} \frac{2m}{z^3}$

• Direction: From south to north pole of the dipole.

Ampère’s Law and Solenoids

Ampère’s Law

Ampère’s law relates the magnetic field around a closed loop to the current passing through the loop.

• Ampère’s Law: $\oint \vec{B} \cdot d\vec{s} = \mu_0 I_{\text{through}}$

• Solenoid Magnetic Field: $B = \mu_0 n I$ (where $n$ is windings per unit length)

Magnetic Forces

Force on Moving Charges

A charged particle moving in a magnetic field experiences a force perpendicular to both its velocity and the field.

• Magnetic Force: $F_{\text{on } q} = q v B \sin \alpha$

• Vector Form: $\vec{F}_{\text{on } q} = q \vec{v} \times \vec{B}$

• Circular Motion: If velocity is perpendicular to $\vec{B}$, the particle moves in a circle of radius $r_{\text{cyc}} = \frac{m v}{q B}$

Force on Current-Carrying Wires

Wires carrying current in a magnetic field experience a force, which can be calculated using the cross product.

• Force on Wire: $F_{\text{wire}} = I l B \sin \alpha$

• Vector Form: $\vec{F}_{\text{wire}} = I \vec{l} \times \vec{B}$

Electromagnetic Induction

Faraday’s Law and Lenz’s Law

Changing magnetic flux induces an emf in a loop, and the direction of induced current opposes the change in flux (Lenz’s law).

• Faraday’s Law: $\mathcal{E} = -\frac{d\Phi_m}{dt}$

• Lenz’s Law: Induced current creates a magnetic field opposing the change in flux.

• Magnetic Flux: $\Phi_m = B \cdot A \cos \theta$

Motional emf

• Motional emf: $\mathcal{E} = v l B$

Maxwell’s Equations

Summary of Maxwell’s Equations

Maxwell’s equations unify electricity and magnetism, describing how electric and magnetic fields are generated and altered by charges and currents.

• Gauss’s Law: $\oint \vec{E} \cdot d\vec{A} = \frac{Q_{\text{in}}}{\epsilon_0}$

• Gauss’s Law for Magnetism: $\oint \vec{B} \cdot d\vec{A} = 0$

• Faraday’s Law: $\oint \vec{E} \cdot d\vec{s} = -\frac{d\Phi_m}{dt}$

• Ampère-Maxwell Law: $\oint \vec{B} \cdot d\vec{s} = \mu_0 I_{\text{through}} + \mu_0 \epsilon_0 \frac{d\Phi_e}{dt}$

Fundamentals of Circuits

Circuit Elements and Diagrams

Basic circuit elements include batteries, wires, resistors, bulbs, junctions, capacitors, and switches. Circuit diagrams use standardized symbols to represent these elements.

• Batteries: Provide emf by chemical reactions.

• Resistors: Cause energy loss due to collisions.

• Bulbs: Resistors that emit light.

• Capacitors: Store charge separation.

• Kirchoff’s Laws:

  • Junction Rule: $\sum I_{\text{in}} = \sum I_{\text{out}}$

  • Loop Rule: $\sum \Delta V_i = 0$

Table: Comparison of Magnetic Field Formulas

Source	Formula	Key Parameters
Long Straight Wire	$B = \frac{\mu_0}{2\pi} \frac{I}{r}$	Current $I$, distance $r$
Current Loop (center)	$B = \frac{\mu_0}{2} \frac{N I}{R}$	Turns $N$, current $I$, radius $R$
Solenoid	$B = \mu_0 n I$	Windings per length $n$, current $I$
Point Charge	$B = \frac{\mu_0}{4\pi} \frac{q v \sin \theta}{r^2}$	Charge $q$, velocity $v$, angle $\theta$, distance $r$