A circular coil with a radius of 25 cm has 20 turns. The coil is oriented
perpendicularly to a magnetic field whose initial magnitude is 3.2 T.
Suddenly, the magnetic field vanishes in 0.4 s.
a. What is the initial magnetic flux in the coil?
b. What is the induced emf in the coil?
c. If the net resistance of the coil is 6.8 Ù what is the
magnitude of the induced current in the coil?
d. What is the direction of the induced current in the coil?
e. What is the rate of thermal energy generated by the coil?
To solve this problem, we need to use Faraday's Law of electromagnetic induction, which states that the induced electromotive force (emf) in a wire loop is equal to the rate of change of magnetic flux through the loop.
a. To find the initial magnetic flux in the coil, we need to use the formula for the magnetic flux through a coil, which is given by:
Φ = B * A * N
Where:
Φ is the magnetic flux
B is the magnetic field strength
A is the area of the loop
N is the number of turns in the coil
In this case, the magnetic field strength is 3.2 T, the radius of the coil is 25 cm (or 0.25 m), and there are 20 turns in the coil.
So, the area of the loop (A) can be calculated using the formula:
A = π * r^2
Where r is the radius of the coil.
Substituting the values into the formulas, we get:
A = π * (0.25)^2 = 0.196 m^2
Φ = 3.2 * 0.196 * 20 = 12.544 T·m^2
Therefore, the initial magnetic flux in the coil is 12.544 T·m^2.
b. The induced emf in the coil is given by the formula:
emf = -N * ΔΦ/Δt
Where:
emf is the induced electromotive force
N is the number of turns in the coil
ΔΦ is the change in magnetic flux
Δt is the change in time
In this case, the change in magnetic flux (ΔΦ) is equal to the initial magnetic flux, and the change in time (Δt) is 0.4 s.
Substituting the values into the formula, we get:
emf = -20 * (12.544 T·m^2)/(0.4 s) = -313.6 V
Therefore, the induced emf in the coil is -313.6 V.
c. To find the magnitude of the induced current (I) in the coil, we can use Ohm's Law:
I = emf/R
Where:
I is the induced current
emf is the induced electromotive force
R is the resistance of the coil
In this case, the resistance of the coil is given as 6.8 Ω.
Substituting the values into the formula, we get:
I = (-313.6 V)/(6.8 Ω) = -46 A
Therefore, the magnitude of the induced current in the coil is 46 A.
d. The direction of the induced current can be determined by Lenz's Law, which states that the induced current will be in a direction that opposes the change in magnetic flux that caused it. In this case, when the magnetic field vanishes, the change in the magnetic flux is a decrease. Therefore, the induced current will flow in a direction that creates a magnetic field to oppose the decrease in the initial magnetic field. This is known as the right-hand rule, where the induced current flows in the opposite direction to the initial magnetic field.
e. To find the rate of thermal energy generated by the coil, we need to use the formula for power:
P = I^2 * R
Where:
P is the power generated
I is the magnitude of the induced current
R is the resistance of the coil
Substituting the values into the formula, we get:
P = (46 A)^2 * 6.8 Ω = 14,974.4 W
Therefore, the rate of thermal energy generated by the coil is 14,974.4 W.