Maxwell’s Equations

PH 203

Professor Lee Carkner

Lecture 25

Transforming Voltage

We often only have a single source of emf

We need a device to transform the voltage

Note that the flux must be changing, and thus the current must be changing Transformers only work for AC current

Basic Transformer

The emf then only depends on the number of turns in each

= N(/t)

Vp/Vs = Np/Ns

Where p and s are the primary and secondary solenoids

Transformers and Current

If Np > Ns, voltage decreases (is stepped down)

Energy is conserved in a transformer so:

IpVp = IsVs

Decrease V, increase I

Transformer Applications

Voltage is stepped up for transmission Since P = I2R a small current is best for

transmission wires Power pole transformers step the voltage down

for household use to 120 or 240 V

Maxwell’s Equations In 1864 James Clerk Maxwell presented to the Royal

Society a series of equations that unified electricity and magnetism and light

∫ E ds = -dB/dt

∫ B ds = 00(dE/dt) + 0ienc

∫ E dA = qenc/0

Gauss’s Law for Magnetism ∫ B dA = 0

Faraday’s Law

∫ E ds = -dB/dt

A changing magnetic field induces a current Note that for a uniform E over a uniform

path, ∫ E ds = Es

Ampere-Maxwell Law

∫ B ds = 00(dE/dt) + 0ienc

The second term (0ienc) is Ampere’s law

The first term (00(dE/dt)) is Maxwell’s Law of Induction

So the total law means Magnetic fields are produced by changing

electric flux or currents

Displacement Current

We can think of the changing flux term as being like a “virtual current”, called the displacement current, id

id = 0(dE/dt)

∫ B ds = 0id + 0ienc

Displacement Current in Capacitor

So then dE/dt = A dE/dt or

id = 0A(dE/dt) which is equal to the real

current charging the capacitor

Displacement Current and RHR

We can also use the direction of the displacement current and the right hand rule to get the direction of the magnetic field Circular around the capacitor axis Same as the charging current

Gauss’s Law for Electricity

∫ E dA = qenc/0

The amount of electric force depends on the amount and sign of the charge Note that for a uniform E over a

uniform area, ∫ E dA = EA

Gauss’s Law for Magnetism

∫ B dA = 0

The magnetic flux through a surface is always zero Since magnetic fields are

always dipolar

Next Time

Read 32.6-32.11 Problems: Ch 32, P: 32, 37, 44

How would you change R, C and to increase the rms current through a RC circuit?

A) Increase all three

B) Increase R and C, decrease C) Decrease R, increase C and D) Decrease R and , increase C

E) Decrease all three

How would you change R, L and to increase the rms current through a RL circuit?

A) Increase all three

B) Increase R and L, decrease C) Decrease R, increase L and D) Decrease R and , increase L

E) Decrease all three

How would you change R, L, C and to increase the rms current through a RLC circuit?

A) Increase all four

B) Decrease and C, increase R and L

C) Decrease R and L, increase C and D) Decrease R and , increase L and C

E) None of the above would always increase current