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Induction
Submitted by: I.D. 034660811
The problem:
A rod in lenght l is sliding with a constant velocity v on two conducting rails. The system is in
inhomogeneous magnetic field produced by the current I which flows in the infinite wire at the
distance a from the top rail: R = 500mΩ = 0.5Ω, I = 10A, l = 8.67cm = 0.0867m, a = 10.5mm =
0.0105m, v = 5.36 m
s
1. What is the electromagnetic force on the rod εind ?
2. What is the current in the circuit Ic ?
3. What is the rate at which the energy is created
dU
dt ?
4. What is the external force F needed to keep the velocity v constant?
5. What is the rate of the work of the external force
dW
dt ?
6. Would there still be an induction if the rails don’t exist?
The solution:
We denote κ =
µ0
4π
=
1.26·10−6
4π
= 1.0026 · 10−7
1. The electro magnetic induction is given by:
εind = −φ̇
Za+l
2κI
a+l
φ =
(vt + x0 )
dr = (vt + x0 ) ln
2κI
r
a
(1)
(2)
a
where x0 is the initial position of the rod.
a+l
εind = −φ̇ = −v ln
2κI = 2.39 · 10−5 V olt
a
2. The current in the circuit is given by:
2κI
v ln a+l
εind
a
Ic =
=
= 4.78 · 10−5 A
R
R
in the clockwise direction.
(3)
(4)
3. The rate that the energy is created:
2
v ln a+l
2κI
dU
a
P =
= εind · Ic =
= 1.14 · 10−9 watt
dt
R
1
(5)
4. The external force needed to keep the velocity v constant is:
Za+l
Za+l
Za+l
ln
2κI
F =
dF = F =
Ic B(r)dr =
Ic ·
dr =
r
a
a
a+l
a
R
2κI
2
v
= 2.13 · 10−10 N
(6)
a
5. The rate of the work of the external force is:
2
v ln a+l
2κI
F · dr
dW
a
=
=F ·v =
= 1.14 · 10−9 watt
dt
dt
R
(7)
and we see that the answer is identical to the answer of q.3.
6. Even if there are no rails, consider a charge in the rod during its movement. The charge “feels“
the Lorentz force pushing it to the edge of the rod. Since the charges with the different signs move to
the opposite edges of the rod, it creates voltage between them. It is the electromagnetic induction.
From another point of view we can think of some loop with one of its sides coincides with the rod
and moves and the others are fixed. Then the magnetic flux through the loop changes with time
which creates the elecromagnetic induction.
But anyway, since the there is no closed circuit here, there is no current.
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