A 30-turn circular coil of radius 4.00 cm and resistance 1.00 (is placed in a magnetic field directed perpendicular to the plane of the coil. The magnitude of the magnetic field varies in time according to the expression
B = 0.0100t + 0.0400t2 , where t is in seconds and B is in tesla. Calculate the induced emf in the coil at t = 5.00 s (8 marks)
To calculate the induced emf in the coil at t = 5.00 s, we need to use Faraday's law of electromagnetic induction, which states that the induced emf in a coil is equal to the rate of change of magnetic flux through the coil.
The magnetic field expression given is B = 0.0100t + 0.0400t^2, where t is in seconds and B is in tesla. To calculate the rate of change of magnetic flux, we need to find the derivative of this equation with respect to time.
Taking the derivative of B with respect to t:
dB/dt = 0.0100 + 2(0.0400t)
Now let's calculate the magnetic flux through the coil. The magnetic flux (Φ) through a coil of radius R is given by the equation:
Φ = B * A,
where A is the area of the coil.
Given that the radius (R) of the coil is 4.00 cm, we can convert it to meters (0.04 m) and use it to calculate the area of the coil using the equation:
A = π * R^2.
Substituting the radius into the equation, we get:
A = π * (0.04)^2 = 0.0016π m^2.
Now let's substitute the derivative of B and the area of the coil into Faraday's law equation:
emf = -dΦ/dt = - (dB/dt) * A.
Substituting the values we found:
emf = - (0.0100 + 2(0.0400t)) * (0.0016π).
To find the induced emf at t = 5.00 s, substitute t = 5.00 s into the equation:
emf = - (0.0100 + 2(0.0400 * 5.00)) * (0.0016π).
Now you can solve this equation to find the induced emf.