magnetic field along the axis of a solenoid ap physics c montwood high school r. casao
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Magnetic Field Along the Axis of a Solenoid
AP Physics C
Montwood High School
R. Casao
• Consider a solenoid of length L, radius R, turns N, carrying current I.
• We will determine an equation for the magnetic field B at an axial point P inside the solenoid.
• Consider the solenoid as a distribution of current loops.
• The magnetic field for any one loop is:
• The net magnetic field in the solenoid is the sum of the magnetic fields of all the loops.
23
22
2o
x
Rx2
RIμB
• Divide the length of the solenoid into small elements of length dx.– The number of turns in a length dx is:
– The amount of current in an element of length dx is:
– The total current in a length dx is:
dN N NdN = dx
dx L L
dxLN
I
di N I N I= di = dx
dx l l
• The magnetic field contribution dB at point P due to each element dx carrying current di is:
dx
LN
IRx2
RμdB
23
22
2o
2
o3
2 2 2
μ R didB
2 x R
• For each element of length dx along the length of the solenoid, the distance x and the angle change. – The value of R remains constant.
• Express x in terms of the angle and find dx.
2
xtan x R tan
Rdx d(R tan ) R d(tan )
dx R sec d
• Substitute:
dsecR
LN
IRtanR2
RμdB 2
23
22
2o
L1)(tan2
dsecNIRμdB
LRtan2
dsecNIRμdB
23
22
23o
23
222
23o
R
R
LsecR2
dsecNIRμdB
Lsec2
dsecNIRμdB
Lsec2
dsecNIRμdB
sec1tan
L1tan2
dsecNIRμdB
63
23o
326
23o
23
232
23o
22
23
223
2
23o
R
R
R
L2dcosNIμ
dB
Lsec2dNIμ
dB
Lsec2dsecNIμ
dB
LsecR2
dsecNIRμdB
o
o
3
2o
63
23o
• Integrate from 1 to 2:
12o
o
o
o
sinsinL2NIμ
B
sinL2NIμ
B
dcosL2NIμ
B
L2dcosNIμ
dB
2
1
2
1
2
1
2
1
• If point P is at the midpoint of the solenoid and if the solenoid is long in comparison to the radius R, then 1 = -90° and 2 = 90°. The result is the equation for the magnetic field at the center of a solenoid.
LNIμ
B
2L2NIμ
11L2NIμ
B
90sin90sinL2NIμ
B
o
oo
o
• If point P is a point at the end of a long solenoid towards the bottom, then 1 = 0° and 2 = 90°. The answer shows that the magnetic field at the end of a solenoid approaches ½ the value at the center of the solenoid.
L2NIμ
B
1L2NIμ
01L2NIμ
B
0sin90sinL2NIμ
B
o
oo
o
• Graph of magnetic field B at axial points vs. distance x for a solenoid.
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