Induced EMFs and Electric Fields

AP Physics C

Montwood High School

R. Casao

• A changing magnetic flux induced an EMF and a current in a conducting loop.

• An electric field is created in the conductor as a result of the changing magnetic flux.

• The law of electromagnetic induction shows that an electric field is always generated by a changing magnetic flux, even in free space where no charges are present.

• The induced electric field has properties that are very different from those of an electrostatic field produced by stationary charges.

• Consider a conducting loop of radius r in a uniform magnetic field that is perpendicular to the plane of the loop.

• If the magnetic field changes with time, Faraday’s law tells us that an EMF given by

is induced in the loop.

m-dΦEMF =

dt

• The induced current produced implies the presence of an induced electric field E, which must be tangent to the loop since all points on the loop are equivalent.

• The work done in moving a test charge q once around the loop is equal to W = q·EMF.

• The electric force on the charge is F = q·E, the work done by this force in moving the charge around the loop is W = q·E·2·π·r, where 2·π·r is the circumference of the loop.

• The two equations for work are equal to each other: q·EMF = q·E·2·π·r, so

EMFE=

2 π r

• Combining this equation for the electric field, Faraday’s law, and the fact that magnetic flux Φm = B·A = B·π·r2 for a circular loop shows that the induced electric field is:

• The negative sign indicates that the induced electric field E opposes the change in the magnetic field.

2m

2

d B rdΦ-1 -1E = =

2 r dt 2 r dt

- r dB -r dBE= =

2 r dt 2 dt

• An induced electric field is produced by a changing magnetic field even if there is no conductor present.

• A free charge placed in a changing magnetic field will experience an electric field of magnitude:

• The EMF for any closed path can be expressed as the line integral of over the path.

-r dBE =

2 dt

E•ds

• The electric field E may not be constant, and the path may not be a circle, therefore, Faraday’s law of induction can be written as:

• The induced electric field E is a non-conservative, time-varying field that is generated by a changing magnetic field.

• The induced electric field E can’t be an electro-static field because if the field were electrostatic, hence conservative, the line integral of over a closed loop would be zero (dΦm/dt = 0).

m-dΦE•ds=

dt

E•ds

Electric Field Due to a Solenoid

• A long solenoid of radius R has n turns per unit length and carries a time-varying current that varies sinusoidally as , where Io is the maximum current and ω is the angular frequency of the current

source. • A. Determine the electric field outside the solenoid, a distance r from the axis.

oI=I cos ω t

• Take the path for the line integral to be a circle centered on the solenoid.

• By symmetry, the magni-tude of the electric field E is constant and tangent to the loop on every point of radius r.

• The magnetic flux through the solenoid of radius R is:

2mΦ =B A=B R

• Applying Faraday’s law:

m

2

2

B AdΦE•ds= =

dt

d B RE•ds=

dtdB

E•ds= Rdt

d

dt

• The electric field E is constant at all points on the loop:

2

2

2

2

2

dBE•ds= R

dtdB

E ds R ds 2 rdtdB

E 2 r Rdt

R dBE

2 r dt

R dBE

2 r dt

• The magnetic field inside the solenoid is:

B = μo·n·I

• Substituting:

2 2o

2o

2oo

2o o

d μ n IR dB RE

2 r dt 2 r dtd Iμ n R

E2 r dt

d I cos ω tμ n RE

2 r dt

d cos ω tμ n I RE

2 r dt

• The electric field varies sinusoidally with time, and its amplitude fall off as 1/r outside the solenoid.

• B. What is the electric field inside the solenoid, a distance r from its axis?

2o o

2o o

2o o

d ω tμ n I RE sin ω t

2 r dt

μ n I R dtE sin ω t ω

2 r dt

μ n I RE ω sin ω t

2 r

• Inside the solenoid, r < R, the magnetic flux through the integration loop is Φm = B·π·r2.

m

2

2

o2

B AdΦE•ds= =

dt

d B rE•ds=

dtdB

E•ds= rdtd μ n I

E ds rdt

d

dt

2o

o2o

2o o

2o o

2o o

2o o

d IE ds r μ n

dt

d I cos ω tE ds r μ n

dt

d cos ω tE ds r μ I n

dtd ω t

E ds r μ I n sin ω tdt

dtE ds r μ I n sin ω t ω

dt

E ds r μ I n ω sin ω t

• The amplitude of the electric field inside the solenoid increases linearly with r and varies sinusoidally with time.

2o o

2o o

o o

ds 2 r

E 2 r r μ I n ω sin ω t

r μ I n ω sin ω tE

2 rr μ I n ω sin ω t

E2