Maxwell’s Equations (so far…)

Maxwell’s Equations (so far…)

0

inside

q

AdE

0 AdB

enclosed0isdB

0 sdE

*Not complete

*Not complete

sd

E

rEsdE 2

0 sdE

for fields made by charges at rest.

Can a distribution of static charges make this field?

Electrostatic forces are conservative.

The change in potential around a loop must be zero.

0 sdE

means:

No curly electric fields.

BUT: This is only true for “Coulomb” fields (fields caused by stationary charges).

There is another way to make electric fields.

increasing B

E

E

E

E

E

E

E

E

Where there is a time-varying magnetic field, there is also a curly electric field.

increasing i

Curly electric field (both inside and outside solenoid)

E

increasing B

steady i

No curly electric field

changingnot B

We call the curly electric fields Non-Coulomb electric fields ENC

They are related to magnetic fields that are changing in time:

dt

BdENC

increasing i

Which direction does the electric field curl?

dt

Bd

NCE

increasing i

Which direction does the electric field curl?

NCE

dt

Bd

Right thumb along

Fingers curl in direction of

dt

Bd

NCE

increasingB out,

Which direction does the electric field curl?

page into dt

Bd

NCE

B

decreasingB out,

Which direction does the electric field curl?

page ofout dt

Bd

NCE

B

increasingB in,

Which direction does the electric field curl?

page ofout dt

Bd

NCE

B

decreasingB in,

Which direction does the electric field curl?

page into dt

Bd

NCE

B

increasing i

What if we put a conducting wire around the solenoid?

NCE

NCE

A current is induced in the wire.

B

SolenoidB increasing

Metal wire

1i2i

NCE

NCE

NCE

NCE

1r 2r

How big is the current i2?

EMF (ElectroMotive Force)

EMF is actually not a force.

It is the energy per unit charge added to a circuit during a singleround trip.

EMF = sdENC

Units: Volts

B

Metal wire

1i2i

NCE

NCE

NCE

NCE

22 rEsdE NCNC

EMF =

1r 2r

SolenoidB increasing

B

Metal wire

1i2i

NCE

NCE

NCE

NCE

in wire resistance

EMF2 i

1r 2r

(Ohm’s Law)电阻

SolenoidB increasing

1i

We can measure ENC by measuring the induced current.

2i

B

1i2i

1r 2r

Experiments: i2 is only present when i1 is changing.

i

t

1i

2i

EMF dt

dB

B

1i2i

1r 2r

Experiments: i2 is proportional to the area of the solenoid.

B

1i2i1r

2r

21rEMF

Faraday’s Law

21rBdt

d

This is the magnetic flux through the loop.

B

B

1i2i

1r 2rEMF

Faraday’s Law

dt

d B

The EMF around a closed path is equal to the rate of change of the magnetic flux inside the path.

EMF

Faraday’s Law

AdBdt

dsdE

The EMF around a closed path is equal to the rate of change of the magnetic flux inside the path.