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Electricity and magnetism

Electricity Electric charge, electric field and potential, current, resistance and capacitance

MagnetismMoving electric charge, magnetic field, magnetic force on moving charges

Electric charge generates electric field:

Gauss’ Law

Electric field exert forces on a charge inside it:

F = qE

Electric potential and voltage:

E = -grad.U, Voltage = ΔU =

Electric field energy density:

Capacitor stores electric energy:

Electric current generates magnetic field:

Ampere’s Law:

The magnetic field lines are always loops around the electric currents.

Magnetic field exerts forces

on moving charges: Lorentz force: F = qv×B

On a section of a wire: F = IL×B

torque on a loop of a wire: τ = IA×B = μ ×B

Magnetic dipole: μ = IA

The Hall effect: ΔV = BI/(nqw)Electric field moves charges and moving charges form current:

Resistance and resistivity:

,

Circuits, power, battery emf, connection rules (R and C in series and parallel) and Kirchhoff’s rules.

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Electric currents and magnetic fields

Magnets were discovered in ancient times. The Chinese discovered the Earth magnet and invented the compass 220 BC.

2000+ years

Earth's magnetic field (and the surface magnetic field) is approximately a magnetic dipole, with one pole near the north pole (see Magnetic North Pole) and the other near the geographic south pole (see Magnetic South Pole). An imaginary line joining the magnetic poles would be inclined by approximately 11.3° from the planet's axis of rotation. The cause of the field is explained by dynamo theory.

South

North

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Electric currents and magnetic fields

In 1820, Hans Christian Oersted discovered that currents generate magnetic fields:

Direction: Right-hand rule

R

Magnitude:

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Moving charges and magnetic field and magnetic force

Current comes from moving charges. Magnetic field actually relates to moving charge as electric field relates to charge, through the force F on the charge: Electric field E and charge q: F = qE, here the force F is the electric charge q. Magnetic field B and moving charge qv: F = qv×B, here the force F is the electric c

harge q which is moving with velocity v.

(comes from the cross product of two vectors)

Combine the two formulas, one has the famous Lorentz force law:

F = qE + qv×B

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The force between two wires with currents

The B field generated by I1 in wire 1 at the location of wire 2 is

For small charge dq in wire 2, the force it experiences follows Lorentz law: dF = dq v×B

Check the vectors directions, the magnitude dF is

dF = dq vB

dq v = dqdl/dt = dq/dt dl = I2dl

R

IB

2

10

B

B

dq

R

LIIdl

R

II

R

IdlIBvdqF

LLL

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0

21010

0

2

0

So F over L is

The direction of the force F depends on the direction of the two currents: same direction, attract; difference direction: repel.

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The definition of the current unit Ampere and examples

When I1 = I2 = I, and the same direction, then the current I is defined to be 1 Ampere if the two wires are 1 meter apart, and the force between them is 2×10-7 N per 1 meter wire length.

In the diagram, one current flows through the straight wire and another current flows around the wire loop, with the magnitudes shown. What are the magnitude and direction of the force exerted on the wire loop by the current in the straight wire?

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example

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Biot-Savart law

When the current follows a wire that is not straight, we use Biot-Savart law to calculate the B field this current generates at a point P in space:

This is similar to Coulomb’s Law that relates electric field to electric charges:

The B field generated by the full section of wire is then:

The formular relates magnet field to currents.

30

4 r

Idd

r

B

l

3

0

4 r

dI

rB

l

ldBd

r

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and

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Example

Using Biot-Savart Law to derive the formula for magnetic field of a long straight wire:

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magnetic field of a long straight wire

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Example

magnetic field at the center of a half circle:

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Ampere’s Law

Ampere’s Law:

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Examples of using Ampere’s Law

Current genersted magnetic field

Outside a long straight wire Inside the wire Of a sheet conductor Of a solenoid Of a toroid

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Examples of using Ampere’s Law

Current genersted magnetic field

Inside the wire

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Examples of using Ampere’s Law

Current genersted magnetic field

of a sheet conductor

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Examples of using Ampere’s Law

Current genersted magnetic field

Of a solenoid

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Examples of using Ampere’s Law

Current genersted magnetic field

Of a toroid