Magnetic Fields, Forces, and Circuits: Study Notes for PHYS112 Practice Exam

Magnetic Fields and Magnetic Forces

Magnetic Dipole Moment and Torque

The magnetic dipole moment is a vector quantity associated with current loops and magnets, representing the strength and orientation of their magnetic properties. When placed in a magnetic field, a current loop experiences a torque that tends to align its dipole moment with the field.

• Magnetic Dipole Moment ($\vec{m}$): For a current loop, $\vec{m} = I \cdot \vec{A}$, where $I$ is the current and $\vec{A}$ is the area vector perpendicular to the loop.

• Torque on a Current Loop: $\vec{\tau} = \vec{m} \times \vec{B}$, where $\vec{B}$ is the magnetic field.

• Direction of Area Vector: Determined by the right-hand rule: curl fingers in the direction of current, thumb points in the direction of $\vec{A}$.

• Force on a Wire: $\vec{F} = I (\vec{L} \times \vec{B})$, where $\vec{L}$ is the length vector of the wire.

Example: A rectangular loop in a uniform magnetic field experiences a torque that depends on the orientation of the loop relative to the field.

Lorentz Force on Moving Charges

The Lorentz force is the force experienced by a charged particle moving in a magnetic field. It is always perpendicular to both the velocity of the particle and the magnetic field.

• Lorentz Force: $\vec{F} = q (\vec{v} \times \vec{B})$, where $q$ is the charge, $\vec{v}$ is velocity, and $\vec{B}$ is the magnetic field.

• Direction: Use the right-hand rule for positive charges; for electrons (negative charge), the force is in the opposite direction.

• Magnitude: $F = |q| v B \sin\theta$, where $\theta$ is the angle between $\vec{v}$ and $\vec{B}$.

Example: An electron moving perpendicular to a uniform magnetic field will experience a force causing it to move in a circular path.

Magnetic Fields Due to Currents

Magnetic Field Around a Straight Wire

A current-carrying wire produces a magnetic field in the surrounding space. The field lines form concentric circles around the wire.

• Magnetic Field at Distance $r$: $B = \frac{\mu_0 I}{2\pi r}$, where $\mu_0$ is the permeability of free space.

• Direction: Use the right-hand rule: thumb in direction of current, fingers curl in direction of field.

Example: Two parallel wires carrying current in the same direction will attract each other; if currents are opposite, they repel.

Magnetic Field Inside a Solenoid

A solenoid is a coil of wire designed to produce a uniform magnetic field inside its core when current flows through it.

• Magnetic Field Inside: $B = \mu_0 n I$, where $n$ is the number of turns per unit length, $I$ is the current.

• Inductance of a Solenoid: $L = \mu_0 n^2 A l$, where $A$ is the cross-sectional area and $l$ is the length.

Example: A solenoid with 400 turns, length 0.8 m, and radius 2.0 cm carrying 0.5 A current produces a strong uniform field inside.

Electromagnetic Induction

Faraday's Law and Induced EMF

Changing magnetic flux through a circuit induces an electromotive force (EMF), which can drive a current.

• Faraday's Law: $\mathcal{E} = -\frac{d\Phi_B}{dt}$, where $\Phi_B$ is the magnetic flux.

• Magnetic Flux: $\Phi_B = B A \cos\theta$, with $A$ as area and $\theta$ as angle between $\vec{B}$ and normal to the area.

• Lenz's Law: The induced EMF creates a current whose magnetic field opposes the change in flux.

Example: If the magnetic field through a loop increases, the induced current will flow to oppose the increase.

Electric Circuits: Current, Resistance, and Power

Ohm's Law and Power in Circuits

Electric circuits consist of sources of EMF, resistors, and other components. The relationship between voltage, current, and resistance is given by Ohm's Law.

• Ohm's Law: $V = IR$, where $V$ is voltage, $I$ is current, $R$ is resistance.

• Power Dissipated: $P = IV = I^2 R = \frac{V^2}{R}$

• Series and Parallel Resistors:

  • Series: $R_{eq} = R_1 + R_2 + \ldots$

  • Parallel: $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \ldots$

Example: In a circuit with a 9 V battery and a 4.5 $\Omega$ resistor, the current is $I = \frac{9}{4.5} = 2$ A, and the power is $P = 9 \times 2 = 18$ W.

Kirchhoff's Laws

Kirchhoff's laws are used to analyze complex circuits with multiple loops and junctions.

• Kirchhoff's Current Law (KCL): The sum of currents entering a junction equals the sum leaving.

• Kirchhoff's Voltage Law (KVL): The sum of EMFs and potential drops around any closed loop is zero.

Example: In a circuit with two bulbs and a switch, closing the switch changes the current distribution and power in each bulb.

Summary Table: Key Equations and Concepts

Concept	Equation	Description
Magnetic Dipole Moment	$\vec{m} = I \cdot \vec{A}$	Current loop's magnetic strength and orientation
Torque on Loop	$\vec{\tau} = \vec{m} \times \vec{B}$	Rotational effect of magnetic field on loop
Lorentz Force	$\vec{F} = q (\vec{v} \times \vec{B})$	Force on moving charge in magnetic field
Magnetic Field (Wire)	$B = \frac{\mu_0 I}{2\pi r}$	Field at distance $r$ from straight wire
Magnetic Field (Solenoid)	$B = \mu_0 n I$	Field inside a long solenoid
Faraday's Law	$\mathcal{E} = -\frac{d\Phi_B}{dt}$	Induced EMF from changing magnetic flux
Ohm's Law	$V = IR$	Relationship between voltage, current, resistance
Power in Circuit	$P = IV$	Electrical power dissipated by resistor

Additional info: These notes cover topics from chapters on magnetic fields, electromagnetic induction, and electric circuits, corresponding to chapters 19, 20, 21, and 22 in a typical college physics curriculum.