Question:
A circular loop with a radius of 0.2 meters and a resistance of 2 ohms is placed in a magnetic field with a magnitude of 0.5 Tesla. The loop is parallel to the field lines and, at time t = 0, the magnetic field is increasing uniformly at a rate of 0.1 Tesla/second. Calculate the magnitude and direction of the induced current in the loop at this instant.
Answer:
The magnitude of the induced current in the loop can be calculated using Faraday's law of electromagnetic induction. Faraday's law states that the induced EMF (electromotive force) is equal to the rate of change of magnetic flux through the loop. The induced current can then be found by dividing this induced EMF by the resistance of the loop.
The magnetic flux through the loop is given by the product of the magnetic field strength, the area of the loop, and the cosine of the angle between the magnetic field lines and the normal to the loop.
Let's calculate the magnetic flux through the loop:
Given parameters:
The area of the loop is given by the formula: A = π * r^2
So, the area of the loop is: A = π * (0.2 m)^2
The angle between the magnetic field lines and the normal to the loop is 0 degrees since the loop is parallel to the field lines. Therefore, the cosine of the angle is 1.
The magnetic flux through the loop is then: Φ = B * A * cos(0°)
Plugging in the values, we get: Φ = (0.5 T) * (π * (0.2 m)^2) * 1
Simplifying the expression, we find: Φ = 0.0314 T·m^2
Now, let's calculate the induced EMF in the loop:
The induced EMF is given by the rate of change of magnetic flux with time. Since the magnetic field is increasing uniformly at a rate of 0.1 T/s, the rate of change of magnetic flux is equal to the product of the rate of change of magnetic field and the area of the loop.
The rate of change of magnetic flux is then: dΦ/dt = (dB/dt) * A
Substituting the given values, we get: dΦ/dt = (0.1 T/s) * (π * (0.2 m)^2)
Simplifying the expression, we find: dΦ/dt = 0.0126 T·m^2/s
Finally, we can calculate the induced current by dividing the induced EMF by the resistance of the loop:
The induced current is given by: I = (dΦ/dt) / R
Substituting the given value for resistance, we get: I = (0.0126 T·m^2/s) / 2 Ω
Simplifying the expression, we find: I = 0.0063 A
Therefore, the magnitude of the induced current in the loop at this instant is 0.0063 Amperes, flowing in the direction according to the right-hand rule.