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1. Solenoid: The magnetic field inside a very long solenoid must be along the axis of the solenoid. The magnetic field outside the solenoid is approximately zero. A very long solenoid of radius b carries a current I with turns per length. We want to use Ampere's law to find the magnetic field at a point P a distance ? from the central axis. Assume 7

1. Solenoid: The magnetic field inside a very long solenoid must be along the axis of the solenoid. The magnetic field outside the solenoid is approximately zero. A very long solenoid of radius b carries a current I with turns per length. We want to use Ampere's law to find the magnetic field at a point P a distance ? from the central axis. Assume 7

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symmetries. 1. Solenoid: The magnetic field inside a very long solenoid must be along the axis of the solenoid. The magnetic field outside the solenoid is approximately zero. A very long solenoid of radius b carries a current I with turns per length. We want to use Ampere's law to find the magnetic field at a point P a distance ? from the central axis. Assume r<b so point P is inside the solenoid. a. To take advantage of the symmetry, we choose the Amperian loop to be a rectangle. The rectangle must extend outside the solenoid to enclose current. We need to introduce a height H to make a closed loop but H must not be in the final answer. We will calculate $5 ds at one leg at a time. i. What is the integral 5 ds along the left leg of the Amperian loop, i.e., outside the solenoid? ii. What is the integral 5 ds along the top and bottom horizontal legs of the Amperian loop? Hint: think about the angle between 8 and ds along these legs. iii. What is the integral - ds along the right leg of the Amperian loop, i.e., inside the solenoid? Answer in terms of the unknown 8 and H, b and/or 7. b. What is the current going through the rectangle? Answer in terms of I, n, H, b and/or r. B & 0 H I (out) 0000000 B Use your results above and apply Ampere's law to find the magnetic field inside the solenoid. Holl ********. E BR 0 n tums per length 7 (in)
Transcribed Image Text:symmetries. 1. Solenoid: The magnetic field inside a very long solenoid must be along the axis of the solenoid. The magnetic field outside the solenoid is approximately zero. A very long solenoid of radius b carries a current I with turns per length. We want to use Ampere's law to find the magnetic field at a point P a distance ? from the central axis. Assume r<b so point P is inside the solenoid. a. To take advantage of the symmetry, we choose the Amperian loop to be a rectangle. The rectangle must extend outside the solenoid to enclose current. We need to introduce a height H to make a closed loop but H must not be in the final answer. We will calculate $5 ds at one leg at a time. i. What is the integral 5 ds along the left leg of the Amperian loop, i.e., outside the solenoid? ii. What is the integral 5 ds along the top and bottom horizontal legs of the Amperian loop? Hint: think about the angle between 8 and ds along these legs. iii. What is the integral - ds along the right leg of the Amperian loop, i.e., inside the solenoid? Answer in terms of the unknown 8 and H, b and/or 7. b. What is the current going through the rectangle? Answer in terms of I, n, H, b and/or r. B & 0 H I (out) 0000000 B Use your results above and apply Ampere's law to find the magnetic field inside the solenoid. Holl ********. E BR 0 n tums per length 7 (in)
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