Post on 04-Feb-2021
Units of magne-c field: Tesla
1 T = 10000 Gauss
Surface of earth: 0.5×10‐4 T Bar magnet: 1.5 T Strong superconduc-ng magnet: 30 T
dB =
µ0I4π
dS × r̂r2
dS
r
Current
Biot‐Savart Law
dB
µ0 = 4π ×10
−7T im / A
Infinite straight wire example
Z
a
a .
b . c .
d .
e .
The segment of wire (total length = 6R) is formed into the shape shown and carries a current I. What is the magnitude of the
resul-ng magne-c field at the point P?
a.
b.
c.
d.
e.
The segment of wire (total length = 6R) is formed into the shape shown and carries a current I. What is the magnitude of the
resul-ng magne-c field at the point P?
Ques-on 1: Equal currents of magnitude I travel into the page in wires M and N. Eight direc-ons are indicated by leYers A
through H.
The direc-on of the magne-c field at point P is
a. B. b. C. c. D. d. E. e. F.
Solenoid
Z
a
B =µ0I2
a2
a2 + z2( )32
Approximate solenoid as bunch of separate loops
Magne-c field at the center of the solenoid
L/2
L : length of solenoid n: coil density (#/m) a: radius of the coil
Looks like integra-on is required again, but how?
a
Z
dz Current through each coil: I Number of coils in dz: ndz
dB = nµ0I2
a2dz
a2 + z2( )32
B =µ0nI2
a2dz
a2 + z2( )32
−L /2
L /2
∫
B =µ0nIa
2
2L / 2
a2 a2 + L4
2⎛⎝⎜
⎞⎠⎟
12
−−L / 2
a2 a2 + L4
2⎛⎝⎜
⎞⎠⎟
12
⎡
⎣
⎢⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥⎥
Z
dz Current through each coil: I Number of coils in dz: ndz
B =µ0nIa
2
2L / 2
a2 a2 + L4
2⎛⎝⎜
⎞⎠⎟
12
−−L / 2
a2 a2 + L4
2⎛⎝⎜
⎞⎠⎟
12
⎡
⎣
⎢⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥⎥
B =µ0nIa
2
2L
a2 a2 + L4
2⎛⎝⎜
⎞⎠⎟
12
If L>>a
B ~ µ0nIa2
2L
a2 L4
2⎛⎝⎜
⎞⎠⎟
12
=µ0nI2
LL / 2
= µ0nI
Z
dz What about the field at the very end?
B =µ0nI2
a2dz
a2 + z2( )32
0
L
∫
B =µ0nIa
2
2L
a2 a2 + L2( )12
−0
a2 a2( )12
⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥
B =µ0nIa
2
2L
a2 (a2 + L2 )1/2
If L>>a
B ~ µ0nIa2
2L
a2 L2( )12
=µ0nI2
LL=µ0nI2
B µ0nI
Ampere’s Law
Current
Amperian Loop
B ⋅dS = µ0∫ I
Integral evaluated around any closed path where I is the total current passing through any surface defined by the path
Example 1: Inside and outside of a wire
Example 2: Solenoid
d
Magne-c Flux
ΦB =
B ⋅dA∫
From last Thursday: Torque on current loop in magne-c field
µ = IA
µ : magne-c moment
Magne-c field applies torque on magne-c moment
Example: Torque on magne-c moment
µ B
Clicker ques-on #3: Which way will the magne-c field try to orient the moment?
a. Clockwise b. Counter clockwise c. Out of the board d. Into the board