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Ampere’s Law
Ampere’s Law
• Gauss’ law allowed us to find the net electric field due to any charge distribution (with little effort) by applying symmetry.
• Similarly the net magnetic field can be found with little effort if there is symmetry using Ampere’s law.
Ampere’s Law
• Ampere’s law,
• Where the integral is a line integral.• B.ds is integrated around a closed loop
called an Amperian loop.• The current ienc is net current enclosed by
the loop.
encisdB 0.
Ampere’s Law
• ie,
• ie ienc
N
n
nisdB1
0.
N
nni
1
Ampere’s Law
• The figure shows the cross section of 3 arbitrary long straight wires with current as shown.
1i
3i 2i
Ampere’s Law
• Two of the currents are enclosed by an Amperian loop.
1i
3i 2i
Ampere’s Law
• An arbitrary direction for the integration is chosen.
3i
1i
2i
Direction ofintegration
Ampere’s Law
• The loop is broken into elements of length ds (choose in the direction of the integration).
• Direction of B doesn’t need to be known before the integration!
3i
1i
2i
Direction ofintegration
sd
B
Ampere’s Law
• B can be in an arbitrary direction at some angle to ds as shown (from the right hand grip rule we know B must in the plane of page).
• We choose B to be in the
direction as ds.
3i
1i
2i
Direction ofintegration
sd
B
Ampere’s Law
• The right hand screw (grip) rule is used to assign a direction to the enclosed currents.
• A current passing through the loop in the same direction as the thumb are positive ( in the opposite direction -ve).
Ampere’s Law
• Consider the integral,
3i
1i
2i
Direction ofintegration
sd
B
N
n
nisdB1
0. dsB cos
Ampere’s Law
• Applying the screw rule,
3i
1i
2i
Direction ofintegration
sd
B
210cos iidsB
Ampere’s Law
• Example. Find the magnetic field outside a long straight wire with current.
r
I
Ampere’s Law
• We draw an Amperian loop and the direction of integration.
Wire surface
Amperian Loop
Direction ofIntegration
B
sd
0
Ampere’s Law
• Recall,
• Therefore,
• The equation derived earlier.
N
n
nisdB1
0.
rBdsBdsB 2cos IrB 02
r
IB
20
Ampere’s Law
• The positive sign for the current collaborates that the direction of B was correct.
Ampere’s Law
• Example. Magnetic Field inside a Long Straight wire with current.
Wire surface
Amperian Loopr
R
B
sd
Ampere’s Law
• Ampere’s Law,
N
n
nisdB1
0.
rBdsBdsB 2cos
Ampere’s Law
• Ampere’s Law,
• The charge enclosed is proportional to the area encircled by the loop,
N
n
nisdB1
0.
rBdsBdsB 2cos
iR
rienc 2
2
Ampere’s Law
• The current enclosed is positive from the right hand rule.
2
2
02R
rirB
rR
iB
20
2
Applications of Ampere’s Law
Applications of Ampere’s law
• Long-straight wire• Insider a long straight wire• Toroidal coil• Solenoid
Toroidal Coil
Toroidal Coil
r
I0
Ampere Loop, circle radius r
No current flowing through loop thus B = 0 inside the Toroid
Toroid has N loops of wire, carrying a current I0
Toroidal Coil
rI0
Ampere Loop, circle radius r
For each wire going in there is another wire comeing out Thus no nett current flowing through loop thus B = 0 outside the Toroid
Toroidal Coil
rI0
Ampere Loop, circle radius r
For each loop of the coil an extra I0 of current passes through the Ampere Loop
Zoom
B2Circle
r dsB I000NI
Toroid has N loops of wire
r
NIB
2
00
Solenoid
Infinitely Long Solenoid
Zoom looks very similar to the toroid with a very large radius
Wire carrying a current of I0 wrapped around with n coils per unit length
Toroidal Coil: Revisited
r
I0
Central radius R circumference is 2pR
Toroid has N loops of wire, carrying a current I0
Number of coils per unit length n is
R
Nn
2
0000
2nI
R
NIB
From earlier:
Independent of R
Infinitely Long Solenoid
Field at centre is same as torus of infinite radius
Wire carrying a current of I0 wrapped around with n coils per unit length
00nIB
