A coil with 150 turns , a radius of 5.0 cm , and a resistance of 12 ohms surrounds a solenoid with 230 turns/cm,and a radius of 4.5 cm. The current in the solenoid changes at a constant rate from 0 to 2 A in 0.10 sec. Calculate the magnitude and direction of the induced current in the coil.
To calculate the magnitude and direction of the induced current in the coil, you can use Faraday's Law of electromagnetic induction. According to Faraday's Law, the emf (electromotive force) induced in a coil is equal to the rate of change of magnetic flux through the coil.
The formula for the induced emf in a coil is given by:
emf = -N * (dPhi/dt),
where emf is the induced electromotive force, N is the number of turns in the coil, and dPhi/dt is the rate of change of magnetic flux.
To calculate the magnetic flux, you can use the formula for the magnetic field inside a solenoid:
B = u0 * N * I,
where B is the magnetic field, u0 is the permeability of free space (4π * 10^-7 T*m/A), N is the number of turns in the solenoid, and I is the current in the solenoid.
Let's start by calculating the magnetic field within the solenoid.
B = u0 * N * I
= (4π * 10^-7 T*m/A) * (230 turns/cm * 0.045 m/cm) * 2 A
= 0.052 T.
Now, we need to calculate the change of magnetic flux through the coil. The magnetic flux can be calculated using the formula:
Phi = B * A,
where Phi is the magnetic flux and A is the area.
Since the coil surrounds the solenoid, the area of the coil is equal to the area of the solenoid.
A = π * r^2
= π * (0.05 m)^2
= 0.00785 m^2.
Therefore, the magnetic flux through the coil is:
Phi = B * A
= 0.052 T * 0.00785 m^2
= 0.00040782 Wb.
Next, we can calculate the rate of change of magnetic flux:
dPhi/dt = (Phi_final - Phi_initial) / dt,
where Phi_final is the final magnetic flux, Phi_initial is the initial magnetic flux (which is zero in this case since the current in the solenoid starts from 0 A), and dt is the change in time.
dPhi/dt = (0.00040782 Wb - 0 Wb) / 0.1 s
= 0.0040782 Wb/s.
Now, we can calculate the induced emf in the coil using Faraday's Law:
emf = -N * (dPhi/dt)
= -150 * 0.0040782 Wb/s
= -0.6117 V.
Since the emf is negative, it means that the induced current will flow in the opposite direction to the change in magnetic field.
Finally, we can calculate the magnitude and direction of the induced current using Ohm's Law (V = I * R), where V is the voltage across the coil, I is the induced current, and R is the resistance of the coil.
V = emf = I * R.
Rearranging the formula, we get:
I = emf / R
= -0.6117 V / 12 Ω
= -0.051 A.
Therefore, the magnitude of the induced current in the coil is 0.051 A, and it flows in the opposite direction to the change in magnetic field.