Problem 3 Two long solenoids are arranged coaxially (one inside the other, centered on the same axis). Both have length L = 1.0m. The inner solenoid has r₁ = 1.6 cm, n₁ = 220 turns per centimeter, and resistance R₁ = 100. The outer solenoid has radius r2 = 1.8 cm, n₂ = 120 turns per centimeter, and resistance R₂ = 5.3. Since the length of the solenoids is much longer than their diameter, we can ignore the fringing of the magnetic field at the ends and treat the solenoids as infinitely long when calculating the magnetic fields (i.e., we can assume that the magnetic field inside is uniform along the entire length). (a) If the inner solenoid initially carries a current I = 1.5 A that then drops to zero at a steady rate over a time interval At = 25 ms, and if the outer inner solenoid is not connected to any battery, what is the current induced in the outer solenoid during this time interval? You may use the Ampere's law result (from the magnetism chapter) for the field inside an infinite coil to determine the field due to the inner solenoid. (b) If instead it is the outer solenoid that initially has a current I = 1.5 A that then drops to zero at a steady rate over a time interval At = 25 ms,, and the inner solenoid is not connected to any battery, what is the current induced in the inner solenoid during this time interval? Compare your answers to (a) and (b), both numerically and in terms of the mathematical expressions. (c) Will the directions of the currents (and the induced electric fields) be the same in both solenoids, or will they be in opposite directions? Explain.
Problem 3 Two long solenoids are arranged coaxially (one inside the other, centered on the same axis). Both have length L = 1.0m. The inner solenoid has r₁ = 1.6 cm, n₁ = 220 turns per centimeter, and resistance R₁ = 100. The outer solenoid has radius r2 = 1.8 cm, n₂ = 120 turns per centimeter, and resistance R₂ = 5.3. Since the length of the solenoids is much longer than their diameter, we can ignore the fringing of the magnetic field at the ends and treat the solenoids as infinitely long when calculating the magnetic fields (i.e., we can assume that the magnetic field inside is uniform along the entire length). (a) If the inner solenoid initially carries a current I = 1.5 A that then drops to zero at a steady rate over a time interval At = 25 ms, and if the outer inner solenoid is not connected to any battery, what is the current induced in the outer solenoid during this time interval? You may use the Ampere's law result (from the magnetism chapter) for the field inside an infinite coil to determine the field due to the inner solenoid. (b) If instead it is the outer solenoid that initially has a current I = 1.5 A that then drops to zero at a steady rate over a time interval At = 25 ms,, and the inner solenoid is not connected to any battery, what is the current induced in the inner solenoid during this time interval? Compare your answers to (a) and (b), both numerically and in terms of the mathematical expressions. (c) Will the directions of the currents (and the induced electric fields) be the same in both solenoids, or will they be in opposite directions? Explain.
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Transcribed Image Text:Problem 3
Two long solenoids are arranged coaxially (one inside the other, centered on the same axis). Both
have length L = 1.0m. The inner solenoid has r₁ = 1.6 cm, n₁ = 220 turns per centimeter, and
resistance R₁ = 100. The outer solenoid has radius r2 = 1.8 cm, n₂ = 120 turns per centimeter,
and resistance R₂ = 5.3. Since the length of the solenoids is much longer than their diameter, we
can ignore the fringing of the magnetic field at the ends and treat the solenoids as infinitely long
when calculating the magnetic fields (i.e., we can assume that the magnetic field inside is uniform
along the entire length).
(a) If the inner solenoid initially carries a current I = 1.5 A that then drops to zero at a steady
rate over a time interval At = 25 ms, and if the outer inner solenoid is not connected to any
battery, what is the current induced in the outer solenoid during this time interval? You may
use the Ampere's law result (from the magnetism chapter) for the field inside an infinite coil to
determine the field due to the inner solenoid.
(b) If instead it is the outer solenoid that initially has a current I = 1.5 A that then drops to zero
at a steady rate over a time interval At = 25 ms,, and the inner solenoid is not connected to any
battery, what is the current induced in the inner solenoid during this time interval? Compare
your answers to (a) and (b), both numerically and in terms of the mathematical expressions.
(c) Will the directions of the currents (and the induced electric fields) be the same in both solenoids,
or will they be in opposite directions? Explain.
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