A 22 turn circular coil of radius 5.00 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 teslas. Calculate the magnitude of the induced emf in the coil at t = 4.00 s.
answer in mV.
I know this uses Faraday's equation but I don't know how to properly use the formula. Any help would be awesome!
To calculate the magnitude of the induced electromotive force (emf) in the coil at a given time, we can use Faraday's law of electromagnetic induction. According to Faraday's law, the emf induced in a coil is equal to the rate of change of magnetic flux through the coil.
The magnetic flux through the coil can be determined as the product of the magnetic field (B) and the area (A) of the coil, given by the formula:
Φ = B * A
In this case, the coil is circular with a radius of 5.00 cm (or 0.05 m), so the area can be calculated as:
A = π * r^2
= π * (0.05)^2
= 0.00785 m^2
Next, we need to find the rate of change of magnetic flux with respect to time (dΦ/dt). In order to do this, we differentiate the expression for magnetic field with respect to time:
B = 0.0100t + 0.0400t^2
Taking the derivative of B with respect to t, we get:
dB/dt = 0.0100 + 0.0800t
Now that we have the derivative of magnetic field with respect to time, we can calculate the rate of change of magnetic flux:
dΦ/dt = B * dA/dt
Since the area of the coil is constant, dA/dt = 0. Therefore, we can simplify the equation to:
dΦ/dt = B * 0
= 0
As a result, the rate of change of magnetic flux is 0, which means there is no change in magnetic flux and no induced emf in the coil.
Therefore, the magnitude of the induced emf in the coil at t = 4.00 s is 0 mV.
To calculate the magnitude of the induced emf in the coil at a given time, we can use Faraday's law of electromagnetic induction. The formula for the induced emf is:
emf = -N * dΦ/dt
Where:
emf is the induced electromotive force (in volts)
N is the number of turns in the coil
dΦ/dt is the rate of change of magnetic flux
In this case, we have a circular coil with 22 turns and a changing magnetic field given by B = 0.0100t + 0.0400t^2.
To calculate the magnetic flux, we can use the formula for the magnetic flux through a coil:
Φ = B * A
Where:
Φ is the magnetic flux (in webers)
B is the magnetic field (in teslas)
A is the area of the coil (in square meters)
First, let's find the area of the coil. The radius is given as 5.00 cm, so we can calculate the area as:
A = π * r^2
A = π * (0.0500 m)^2
Now we can calculate the magnetic flux at time t = 4.00 s:
Φ = B * A
Φ = (0.0100t + 0.0400t^2) * π * (0.0500 m)^2
Φ = (0.0100(4.00 s) + 0.0400(4.00 s)^2) * π * (0.0500 m)^2
To find the rate of change of magnetic flux, we differentiate the above equation with respect to time (t). After taking the derivative, substitute t = 4.00 s, then multiply the result by -N (number of turns).
Finally, we multiply the calculated result by 1000 to convert the emf from volts to millivolts, since the answer is required in mV.
Let's perform the calculations step by step:
1. Calculate the area of the coil:
A = π * (0.0500 m)^2
2. Calculate the magnetic flux at t = 4.00 s:
Φ = (0.0100(4.00 s) + 0.0400(4.00 s)^2) * π * (0.0500 m)^2
3. Find the rate of change of magnetic flux by differentiating Φ with respect to t, then substitute t = 4.00 s.
4. Multiply the result by -N (number of turns).
5. Multiply the emf by 1000 to convert it to millivolts.
Performing these calculations will provide the magnitude of the induced emf in the coil at t = 4.00 s in mV.