Induced EMFs and Electric Fields
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Transcript of Induced EMFs and Electric Fields
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