A circular loop of wire (radius = 6.0 cm, resistance = 40 miliamps) is placed in a uniform magnetic field that makes and angle of 30 degrees with the plane of the loop. The magnitude of the field changes with time according to B = 30 sin (20t)mT, where t is measured in s. Determine the magnitude of the current induced in the loop at t = pi/20 s
To determine the magnitude of the current induced in the loop at a given time, we can use Faraday's Law of electromagnetic induction.
The formula for the induced current is given by:
ε = -N * dΦ/dt
Where:
- ε is the induced electromotive force (emf)
- N is the number of turns in the loop
- dΦ/dt is the rate of change of the magnetic flux through the loop
To find the current, we need to calculate the emf and then divide it by the resistance of the loop.
Step 1: Calculate the emf (ε)
The emf can be found by multiplying the rate of change of magnetic flux with the number of turns in the coil.
Φ = magnetic flux through the loop
The magnetic flux (Φ) is given by:
Φ = B * A * cos(θ)
Where:
- B is the magnetic field strength
- A is the area of the loop
- θ is the angle between the magnetic field and the plane of the loop
Given:
- B = 30sin(20t) mT
- A = πr^2 (circular loop)
- r = 6.0 cm = 0.06 m
- θ = 30 degrees = π/6 radians
Substituting the values into the formula, we have:
Φ = (30sin(20t)) * (π(0.06)^2) * cos(π/6)
Step 2: Calculate the rate of change of magnetic flux (dΦ/dt)
To find the rate of change of magnetic flux, we differentiate the magnetic flux equation with respect to time.
dΦ/dt = (d/dt)[(30sin(20t)) * (π(0.06)^2) * cos(π/6)]
Differentiating each term separately:
dΦ/dt = (d/dt)[30sin(20t)] * (π(0.06)^2) * cos(π/6) + (30sin(20t)) * (d/dt)[(π(0.06)^2) * cos(π/6)]
Taking the derivatives:
dΦ/dt = (30 * 20cos(20t) * π(0.06)^2) * cos(π/6) - (30sin(20t)) * (π(0.06)^2) * (-(π/6))sin(20t)
Step 3: Calculate the induced emf (ε)
Now that we have the expression for dΦ/dt, we can substitute it back into the emf equation.
ε = -N * dΦ/dt
Given:
- N = 1 (single loop)
Substituting N and dΦ/dt into the equation:
ε = -1 * [(30 * 20cos(20t) * π(0.06)^2) * cos(π/6) - (30sin(20t)) * (π(0.06)^2) * (-(π/6))sin(20t)]
Step 4: Calculate the induced current (I)
Finally, we can calculate the induced current in the loop by dividing the emf (ε) by the resistance (R) of the loop.
Given:
- R = 40 milliamps = 40 * 10^-3 Ω
Substituting ε and R into the equation:
I = ε / R
Now, substitute t = π/20 s into the equation and calculate the magnitude of the current induced in the loop.