A solenoid has N=1760.0 turns, length d=40 cm , and radius b=0.4 cm, (b<<d) . The solenoid is connected via a switch, S1 , to an ideal voltage source with electromotive force ϵ=7 V and a resistor with resistance R=25 Ohm . Assume all the self-inductance in the circuit is due to the solenoid. At time t=0 , S1 is closed while S2 remains open.

(a) When a current I=0.112 A is flowing through the outer loop of the circuit (i.e. S1 is still closed and S2 is still open), what is the magnitude of the magnetic field inside the solenoid (in Tesla)?

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(b) What is the self-inductance L of the solenoid (in H)?

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(c) What is the current (in A) in the circuit a very long time (t>>L/R) after S1 is closed? .

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(d) How much energy (in J) is stored in the magnetic field of the coil a very long time (t>>L/R) after S1 is closed?

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For the next part, assume that a very long time (t>>L/R) after the switch S1 was closed, the voltage source is disconnected from the circuit by opening switch S1. Simultaneously, the solenoid is connected to a capacitor of capacitance C=751 μF by closing switch S2. Assume there is negligible resistance in this new circuit.

(e) What is the maximum amount of charge (in Coulombs) that will appear on the capacitor?

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(f) How long does it take (in s) after S1 is opened and S2 is closed before the capacitor first reaches its maximum charge?

To answer these questions, we need to use the formulas related to solenoids and circuits. I'll break down the steps for each question:

(a) To find the magnetic field inside the solenoid, we can use the formula for the magnetic field inside a solenoid:

B = μ₀ * N * I / L

Where:
B is the magnetic field (in Tesla),
μ₀ is the permeability of free space (4π * 10^(-7) T*m/A),
N is the number of turns of the solenoid,
I is the current flowing through the solenoid, and
L is the length of the solenoid.

Substituting the given values into the formula, we have:

B = (4π * 10^(-7) T*m/A) * 1760.0 * 0.112 A / 0.4 m

Simplifying the expression will give us the magnitude of the magnetic field inside the solenoid in Tesla.

(b) The self-inductance L of the solenoid can be calculated using the formula:

L = μ₀ * N^2 * A / l

Where:
A is the cross-sectional area of the solenoid, and
l is the length of the solenoid.

The cross-sectional area of the solenoid can be calculated approximately as:

A = π * (b^2)

Substituting the given values into the formulas, we can calculate the self-inductance L in Henrys.

(c) To find the current in the circuit a very long time after S1 is closed, we can use the formula for the steady-state current:

Iss = ε / R

Where:
Iss is the steady-state current,
ε is the electromotive force (voltage) of the ideal voltage source, and
R is the resistance in the circuit.

Substituting the given values into the formula will give us the current in Amperes.

(d) The energy stored in the magnetic field of the coil a very long time after S1 is closed can be calculated using the formula:

U = (1/2) * L * I^2

Where:
U is the energy stored in the magnetic field,
L is the self-inductance of the coil, and
I is the current flowing through the solenoid.

Substituting the given values into the formula will give us the energy stored in Joules.

(e) To find the maximum amount of charge on the capacitor, we can use the formula:

Q = C * V

Where:
Q is the charge on the capacitor,
C is the capacitance of the capacitor, and
V is the voltage across the capacitor.

Substituting the given values into the formula, we can calculate the maximum charge on the capacitor in Coulombs.

(f) The time it takes for the capacitor to reach its maximum charge after S1 is opened and S2 is closed can be calculated using the formula:

τ = R * C

Where:
τ is the time constant of the RC circuit,
R is the resistance in the circuit, and
C is the capacitance of the capacitor.

Substituting the given values into the formula will give us the time it takes in seconds.