Electricity

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Electricity Magnetism

• Seb Oliver
• Lecture 16
• Amperes Law

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Summary Lectures 14 15

• Biot-Savart Law
• (Field produced by wires)
• Centre of a wire loop radius R
• Centre of a tight Wire Coil with N turns
• Distance a from long straight wire
• Force between two wires
• Definition of Ampere

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Other examples of Magnetic field

• Centre of a wire loop radius R
• Centre of a tight Wire Coil with N turns
• Axis of solenoid n turns per unit length
• Distance a from long straight wire

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Another Right-Hand Rule
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Amperes Law
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Magnetic Field from a long wire
Using Biot-Savart Law
r
I
Take a short vector on a circle, ds
B
ds
Thus the dot product of B the short vector ds
is
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Sum B.ds around a circular path
r
I
B
Sum this around the whole ring
ds
Circumference of circle
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Sum B.ds around any path
N.B. this does not depend on r
In fact it does not depend on path
Amperes Law
on any closed loop
where I is the current flowing through the loop
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Amperes Law
Sign comes from direction of loop, current
right hand rule
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Amperes Law
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Amperes Law

• No Different Physics from Biot-Savart Law
• Useful in cases where there is a high degree of
  symmetry
• C.f. Coulombs Law and Gausss Law in
  electrostatics

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Quiz
Currents of 1A, 5A, 2A, flowing in 3 wires as
shown
1A
What is ?B.ds through loops a, b, c, d?
5A
2A
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Examples of using Amperes Law
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Examples

• Long-straight wire
• Insider a long straight wire
• Toroidal coil
• Solenoid

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Long Wire
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Magnetic Field from a long wire
By symmetry
r
Amperes Law on Loop 2
I
B(r1)
Loop1
Br2
Br1
B(r1)
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Magnetic Field from a long wire
r
I
For any closed Ampere Loop the radial components
will always cancel out
By symmetry
Br3
Br4
Loop 2
Loop 3
Br2
Br1
Br1
Thus there is no way to balance a current by a
radial component or produce a radial component
from a current
Br2
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Magnetic Field from a long wire
Tangential component
Take a circle of radius r as the Ampere Loop
r
I
Tangential component
By symmetry at constant r
L.H.S.
or
L.H.S. R.H.S
Q.E.D.
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Inside a Wire
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Inside a wire current carrying I0
We Take our Ampere loop to be a circle of radius r
Assuming that the current density is uniform then
the current flowing through the loop is
Now same as before
A
r
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Field from a long wire
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Toroidal Coil
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Toroidal Coil
I0
Toroid has N loops of wire, carrying a current I0
Ampere Loop, circle radius r
No current flowing through loop thus B 0 inside
the Toroid
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Toroidal Coil
Ampere Loop, circle radius r
I0
For each wire going in there is another wire
comeing out Thus no nett current flowing through
loop thus B 0 outside the Toroid
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Toroidal Coil
I0
Zoom
Toroid has N loops of wire
For each loop of the coil an extra I0 of current
passes through the Ampere Loop
Ampere Loop, circle radius r
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Solenoid
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Infinitely Long Solenoid
Wire carrying a current of I0 wrapped around with
n coils per unit length
Zoom looks very similar to the toroid with a very
large radius
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Toroidal Coil Revisited
I0
Toroid has N loops of wire, carrying a current I0
Central radius R circumference is 2pR
Number of coils per unit length n is
From earlier
Independent of R
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Infinitely Long Solenoid
Wire carrying a current of I0 wrapped around with
n coils per unit length
Field at centre is same as torus of infinite
radius
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Summary Lecture 16

• Amperes Law
• Easier to use than Biot-Savart Law in many cases
• Examples
• Long-wire
• Inside wire
• Toroid
• Solenoid