Magnetic Flux and
Electromagnetic Induction
With Non-Uniform
Magnetic Fields
Magnetic Flux
in a Nonuniform Field
So far, we have assumed that the loop
is in a uniform field. What if that is not
the case?
The solution is to break up the area
into infinitesimal pieces, each so small
that the field within it is essentially
constant. Then:
r r
d Φ m = B ⋅ dA
Φm =
∫
r r
B ⋅ dA
area of loop
2
Example: Magnetic Flux from
a Long Straight Wire
The near edge of a 1.0 cm x 4.0 cm
rectangular loop is 1.0 cm from a long straight
wire that carries a current of 1.0 A, as shown in
the figure.
What is the magnetic flux through the loop?
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Electromagnetic Waves
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Maxwell’s Theory
Maxwell produced a mathematical
formulation of Faraday’s lines of force picture.
He reasoned from this that if a changing
magnetic field produces an electric field, then a
changing electric field should be equivalent to a
current in producing a magnetic field.
Otherwise, there is a paradox. An Amperian
loop near a charging capacitor will predict a
different magnetic field, depending on whether
the surface enclosed by the loop passes through
the current (a) or through the
capacitor gap (b). If the changing electric field
is effectively a current (called the
“displacement current”) there is no paradox.
James Clerk Maxwell
(1831-1879)
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Electromagnetic Waves
Maxwell’s formulation of electricity and
magnetism has an interesting consequence. The
equations can be manipulated to give a wave
equations for E and B of the form:
d 2E
d 2E
= µ 0ε 0 2
2
dx
dt
This can be recognized as describing an
electromagnetic wave traveling through space with a
velocity of:
vEM wave
(4π × 9.0 × 109 Nm 2 /C2 )
8
=
=
=
3.0
×
10
m/s
2
−7
(4π × 10 N/A )
µ0ε 0
1
This is quite a remarkable result. Somehow, equations for charges and currents
making stationary electric and magnetic fields are telling us about electromagnetic waves
traveling through space at the speed of light!
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