Electricity and Magnetism: Magnetism

Properties of magnetic field lines

• Field lines point from North to South (magnetic monopoles do not exist).

• Tangent at a point gives the direction of the field at that point.

• Density (spacing) of lines represents magnitude of the field.

• Field lines cannot cross and are continuous

How is a magnetic field created?

By moving charges.

• Electromagnets - current flowing in a loop or coil of wire.

• Permanent magnets - atomic-level current loops (circulating or spinning electrons) can, in some materials, add together to give a net magnetism.

• Quantum spin.

Magnetic force exerted on a charge in a magnetic field

Give by \vec F_B=q\,\vec v \times \vec B.

Use right hand rule: thumb gives force, first finger gives velocity, second finger gives magnetic field.

Motion of a charged particle moving perpendicular to a magnetic field

Particle travels in a circular path.

Equate centripetal force to magnetic force to find radius of the path and frequency of revolutions (cyclotron frequency).

Motion of a charged particle moving at an angle to a magnetic field

Particle travels in a helical path.

Split the velocity into components parallel and perpendicular to the magnetic field: parallel gives uniform motion (no force acts) and perpendicular gives circular motion.

Lorentz force

The total force \vec F=q\vec E +q\,\vec v \times \vec B acting on a charged particle in both magnetic and electric fields.

Hall effect

When a current-carrying conductor is placed in a magnetic field \vec B, an electric field \vec E_H is generated that is perpendicular to both B and v_d.

This field arises because the magnetic force pushes the charges to the edges of the conductor, resulting in an accumulation of charge and hence a p.d. called Hall voltage, ∆V_H.

Equation for magnetic force on a current-carrying conductor

Force on a wire with endpoints at a and b is

\vec F_B=I\int_a^b d\vec s\times \vec B

If the magnetic field is uniform, it can be taken outside the integration, and if the wire forms a closed loop, \vec F_B=0.

Torque on a rectangular current loop

Given by \vec \tau=I\vec A\times \vec B, where \vec A is the area of the loop

Potential energy due to torque

Torque is the gradient of potential energy, so \frac{dU}{d\theta}=-\tau=-IAB\sin\theta.

Integrate and set U=0 for \theta=\pi/2 to find

U=-IAB\cos\theta=I\vec A \cdot \vec B

Magnetic dipole moment

Of a small coil of area \vec A carrying a current I is \vec m=I\vec A.

Can write potential energy and torque in terms of \vec m as U=-\vec m\cdot\vec B,\,\,\,\, \vec\tau =\vec m \times\vec B.

Biot-Savart law

Given by

d\vec B=\frac{\mu_0 I}{4\pi}\frac{d\vec s \times \hat r}{r^2}

and is used to find the total magnetic field at a point due to a wire.

Magnetic force between two parallel wires

Two current carrying wires will exert magnetic forces on each other. Wire 2 creates a field \vec B_2 at wire 1, which exerts a force \vec F_1 = I_1\vec l × \vec B_2.

If the currents are in the same direction, the wires attract, otherwise they repel.

Ampère’s law

Used to find magnetic field generated by a current, given by

\oint\vec B \cdot d\vec s=\mu_0 I

where I is the total current passing through any surface bounded a closed path.

Magnetic susceptibility

Gives the induced magnetic field, \vec B_m, in a material by

\vec B_m=\chi_m \vec B_0

where \vec B_0 is the external magnetic field.

Diamagnetism

When -1\le \chi_m<0: a magnet is induced in the material which opposes an external magnetic field.

\vec B=(1+\chi_m)\vec B_0\,<\,\vec B_0

When \chi_m=-1, the Meissner Effect occurs (in superconductors). The magnetic field that would be flowing through the material is expelled, and the field lines instead wrap around it.

Paramagnetism

When 0\le\chi_m«1: atoms have randomly orientated individual permanent magnetic moments, but when an external magnetic field is applied, the atoms align with it.

\vec B=(1+\chi_m)\vec B_0\,>\,\vec B_0

Magnetisation (paramagnets)

Defined by \vec M=\frac{\vec m}{V}.

Ferromagnetism

For \chi_m»1: ferromagnets have a permanent magnetic moment due to aligned spins of unpaired electrons. Alignment of spins is due to the exchange interaction.

Tend to align to minimise energy over small regions called Weiss domains, for which adjacent domains are randomly orientated.

Mass spectrometer

Contains a velocity selector and a region with a uniform magnetic field after the velocity selector. The radius of particles in the second magnetic field depends on their mass. Can find the charge to mass ratio of particles.

Velocity selector

Region of perpendicular uniform magnetic and electric fields. The force due to both fields must be equal on the particle for it to travel in a straight line and leave the velocity selector.

Equations for the Hall effect

Hall voltage given by

\Delta V_H=R_H\frac{IB}{t}

where R_H=1/nq is the Hall coefficient and t is the thickness of the conductor.

Found by using the induced electric field in the conductor: E_{H}=\Delta V_{H}/d, where d is the width of the conductor, and equating the electric and magnetic forces on the conductor (conductor is in equilibrium).

Permeability of free space

A measure of the amount of resistance encountered when forming a magnetic field in a vacuum, symbol \mu_0.

Magnetic field in a solenoid

Magnetic field adds for each loop to create a magnetic field that points straight through the middle of the solenoid, but cancels to give zero outside of the solenoid.

Maximum magnetisation

Occurs when all magnetic moments of the atoms are lined up with the external field,

\vec M_{max}=\frac{N\vec\mu}{V}

where N is the number of atoms within the volume V, and \vec \mu is the magnetic moment of an individual atom.

Pierre Curie’s law

In paramagnetic materials, the magnetisation \vec M is proportional to the applied field \vec B_0 and inverseley proportional to the temperature T,

\vec M=C\frac{\vec B_0}{T}

for small values of \vec B_0/T.

Hard vs soft ferromagnets

A hard ferromagnet will retain magnetisation by an external field, i.e., the Weiss domains remain aligned, after the removal of the field, but a soft ferromagnet will not.

How to remove hard ferromagnetisation.

By heating or impact.

At the Curie temperature, the exchange interaction stops, and the material completely loses its permanent magnetic moment and behaves like a paramagnet.

Magnetic flux

Given by \Phi_B=\int_S \vec B \cdot \mathrm{d}\vec a.

Gauss’ law for magnetism

\vec \nabla \cdot\vec B=0

Via \Phi_B=\int_S \vec B \cdot d\vec a=\int_V \vec \nabla\cdot \vec B\, \,\mathrm{d}V=0, which implies that there are no magnetic monopoles (same number of flux lines leave a volume as they do enter it).

Stoke’s theorem

The integral around a closed path, C, of a vector quantity is equal to the integral of the curl of that quantity integrated over any surface S enclosed by the path.

\int_C \vec F\cdot d\vec s=\int_S \vec \nabla \times \vec F \cdot d\vec a

Displacement current

When an electric field varies with time, Ampère’s law no longer holds. This is solved by adding a displacement current

I_d=\epsilon_0\frac{\mathrm{d}\Phi_E}{\mathrm{d}t}

to modify Ampère’s law.

Ampère-Maxwell law

\oint \vec B\cdot \mathrm{d}\vec s=\mu_0I+\mu_0\epsilon_0\frac{\mathrm{d}\Phi_E}{\mathrm{d}t}

This implies that magnetic fields can be produced by both time-varying electric fields and conduction currents.

Differential form of the Ampère-Maxwell law

Found via applying Stoke’s theorem to the Ampère-Maxwell law:

\vec \nabla \times \vec B=\mu_0 J+\mu_0\epsilon_0\frac{\partial \vec E}{\partial t}

Faraday’s law of induction

\varepsilon=\oint \vec E \cdot \mathrm{d}\vec s=-\frac{\mathrm{d}\Phi_B}{\mathrm{d}t}

Which electric fields are conservative?

Those for which the integral of \vec E around a closed path is zero (zero curl).

The electric field from static charges is conservative as no work is done to move a charged particle around a closed path.

Back emf in terms of inductance

\varepsilon_L=-L\frac{\mathrm{d}I}{\mathrm{d}t}

Lenz’s law

An induced electric current flows in a direction such that the current opposes changes in the magnetic field that induced it.

Units of inductance

Henry, H, where 1 H = 1 VsA^-1.

Overdamped (in an RLC circuit)

When overdamped, there are no oscillations in the charged Q stored on the capacitor, and the energy decays rapidly.

Occurs when R>R_c=\sqrt{4L/C}.

Underdamped (in an RLC circuit)

When underdamped, the charge Q stored on the capacitor oscillates, where the amplitude slowly decreases until all energy is dissipated.

How to find energy stored in an inductor.

Find the rate at which energy is stored in the inductor by applying Kirchhoff’s loop rule and multiplying by current. Then integrate to find

U=\frac{1}2 LI²

Rate of energy dissipation in an LC circuit.

No resistor, therefore no energy dissipated as heat.

\frac{dU}{dt}=0

Rate of energy dissipation in an RLC circuit.

Energy is dissipated as heat through the resistor at a rate:

\frac{dU}{dt}=I²R

AC induced emf and current

\varepsilon=\varepsilon_\text{max}\sin(\omega_d t)

i=I\sin(\omega t-\phi)

AC emf and frequency in the UK

emf is 230V and frequency is 50Hz.

How to generate an AC current

• Turn a loop of wire in a magnetic field, or turn a magnet inside a loop of wire.

• Electricity is generated in the wire, as kinetic energy is transformed into electrical energy.

How a motor works

• When a loop of current-carrying wire is placed in a magnetic field, the loop will experience a force upwards on one side and downwards on the other.

• The loop will then turn, with a direction of turn given by the right hand rule.

• Here, electrical energy is converted into kinetic energy.

Phasor diagram

Length indicates magnitude. The phasor rotates at angular speed \omega. The angle between phasors is the phase difference \phi.

Instantaneous voltage over a resistor

v_R=V_R\sin(\omega_d t)

Instantaneous voltage over a capacitor

v_c=V_C\sin(\omega_d t)

Relation between amplitudes in a resistor AC circuit

V_R=I_RR

And v_R and i_R are in phase.

Relation between impedance, resistance and reactance

Z=\sqrt{R²+(X_L-X_C)²}

Reactance

Refers to how current is slowed down by a capacitor (capacitive reactance X_C) or an inductor (inductive reactance X_L).

Units of ohms.

Capacitive reactance

Given by X_C=\frac{1}{\omega_dC}

Inductive reactance

Given by X_L=\omega_dL

Relation between amplitudes in an inductor AC circuit

V_L=I_LX_L

And current i_C lags behind v_C by \pi/2 .

Therefore \phi=\pi/2.

Relation between amplitudes in an capacitor AC circuit

V_C=I_CX_C

And current i_C leads voltage v_C by \pi/2 .

Therefore \phi=-\pi/2.

Equation for instantaneous emf in an RLC circuit

\varepsilon=v_R+v_C+v_L

Energy stored in an LC circuit

1. When the capacitor is fully charged, all energy is stored as electrical energy, and current is zero.

2. As capacitor starts to discharge, current begins to flow and inductor begins to store energy. (Energy shared between inductor and capacitor.)

3. When the capacitor has fully discharged, current is at a maximum, and all energy is stored in the inductor as magnetic energy.

4. Capacitor begins to charge again on the opposite plate to last time, so current begins to decrease, and once again energy is shared between the capacitor and inductor.

5. Capacitor charges again, where electrons are stored on the opposite plate as last time, and cycle repeats.

Transformer equation

\varepsilon_s=\frac{N_s}{N_p}\varepsilon_p