Maxwell`s Equations

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Transcript Maxwell`s Equations

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
e = N(DF/Dt)

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 = -dFB/dt

∫ B ds = m0e0(dFE/dt) + m0ienc

∫ E dA = qenc/e0
Gauss’s Law for Magnetism
∫ B dA = 0
Faraday’s Law

∫ E ds = -dFB/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 = m0e0(dFE/dt) + m0ienc
 The second term (m0ienc) is Ampere’s
law

 The first term (m0e0(dFE/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 = e0(dFE/dt)

∫ B ds = m0id + m0ienc
Displacement Current in Capacitor

So then dFE/dt = A dE/dt or
id = e0A(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/e0

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 w to
increase the rms current through a RC
circuit?
A)
B)
C)
D)
E)
Increase all three
Increase R and C, decrease w
Decrease R, increase C and w
Decrease R and w, increase C
Decrease all three
How would you change R, L and w to
increase the rms current through a RL
circuit?
A)
B)
C)
D)
E)
Increase all three
Increase R and L, decrease w
Decrease R, increase L and w
Decrease R and w, increase L
Decrease all three
How would you change R, L, C and w to
increase the rms current through a RLC
circuit?
A)
B)
C)
D)
E)
Increase all four
Decrease w and C, increase R and L
Decrease R and L, increase C and w
Decrease R and w, increase L and C
None of the above would always
increase current