A 39-turn circular coil of radius 3.20 cm and resistance 1.00 Ω 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 = 0.010 0t + 0.040 0t2, where B is in teslas and t is in seconds. Calculate the induced emf in the coil at t = 5.40 s.

39- turn coil

3.20 cm
1.00 Ω
B= 0.0100t + 0.0400t2
T= 5.40s

lƩl =ΔφB/Δt = N (dB/dt)A


=N[d/dt (0.0100t + 0.0400 t^2)A
=39(0.0100 + 0.0800(5.40))[π(0.0320)^2]
=0.05545
=5.55 x 10^-2 V
=55.5 mV

Oh, magnetic fields and circular coils, this is getting electrifying! At t = 5.40 s, we can calculate the induced emf in the coil using Faraday's law. So, let's get our laugh-inducing math hats on!

Now, Faraday's law states that the induced emf (ε) is equal to the negative rate of change of magnetic flux through the coil. The magnetic flux (Φ) through the coil can be calculated by multiplying the magnetic field (B) by the area of the coil (A). In this case, we have a circular coil, so the area can be calculated using A = π * r^2. And because the magnetic field is changing with time, we also have to multiply by dt to consider the time interval.

So, let's plug everything in and calculate some funny business:

ε = -dΦ/dt = -d(B * A)/dt = -d(B * π * r^2)/dt

Now, let's differentiate the expression for the magnetic field with respect to time:

d(B)/dt = 0.010 0 + 0.040 0 * 2t = 0.010 0 + 0.080 0t

Next, we'll differentiate the expression for the area with respect to time:

d(A)/dt = d(π * r^2)/dt = 2π * r * dr/dt

But since the radius (r) remains constant, dr/dt = 0, so d(A)/dt = 0.

Now we can substitute these derivatives back into the equation:

ε = -d(B * A)/dt = -d(B * π * r^2)/dt = -d(B)/dt * A = -(0.010 0 + 0.080 0t) * A

Now we just need to substitute the values for A and the given time (t = 5.40s):

A = π * r^2 = π * (3.20cm)^2 = π * (3.20 * 10^-2m)^2 = 3.20^2π * 10^-4m^2

ε = -(0.010 0 + 0.080 0t) * A = -(0.010 0 + 0.080 0 * 5.40) * 3.20^2π * 10^-4m^2

Multiplying all those numbers together should give you the induced emf at t = 5.40s. I hope this math-induced humor brightened up your day a bit!

To calculate the induced emf in the coil, we can use Faraday's Law of electromagnetic induction. Faraday's Law states that the induced emf (ε) in a circuit equals the negative rate of change of magnetic flux through the circuit.

The magnetic flux (Φ) through a loop of wire is given by the product of the magnetic field (B) perpendicular to the loop and the area (A) of the loop. In this case, since the coil is circular, the area of the loop is given by A = πr^2.

Now, to find the induced emf at t = 5.40 s, we need to find the rate of change of magnetic flux and then multiply it by the number of turns in the coil (N) to account for all the loops.

Step 1: Find the rate of change of magnetic flux (dΦ/dt):
To find the rate of change of magnetic flux, we differentiate the given expression for the magnetic field with respect to time:
dB/dt = 0.010 0 + 2(0.040 0t) = 0.010 0 + 0.080 0t

Step 2: Calculate the area of the coil:
Given that the radius of the coil is 3.20 cm, the area of the coil is A = π(3.20 cm)^2.

Step 3: Calculate the rate of change of magnetic flux through the coil:
dΦ/dt = B * A
= (0.010 0 + 0.080 0t) * π(3.20 cm)^2

Step 4: Calculate the induced emf (ε):
ε = -N * dΦ/dt
= -39 * (0.010 0 + 0.080 0t) * π(3.20 cm)^2

Step 5: Substitute t = 5.40 s into the equation for ε to find the induced emf at t = 5.40 s.

To calculate the induced emf in the coil, we need to use Faraday's law of electromagnetic induction, which states that the induced emf is equal to the rate of change of magnetic flux through the coil. The magnetic flux through a coil is given by the product of the magnetic field and the area of the coil perpendicular to the magnetic field.

Let's break down the problem step by step:

1. Find the magnetic field at t = 5.40 s:
Plug in t = 5.40 s into the given equation for B:
B = 0.0100t + 0.0400t^2
B = 0.0100(5.40) + 0.0400(5.40)^2
B ≈ 0.0540 + 1.5552
B ≈ 1.6092 T

2. Find the magnetic flux in the coil:
Magnetic flux (Φ) = B * A
The area of the coil (A) can be calculated using the formula for the area of a circle: A = πr^2
A = π(0.0320)^2
A ≈ 0.003213 m^2

Φ = B * A
Φ = 1.6092 * 0.003213
Φ ≈ 0.00517 Wb (webers)

3. Calculate the induced emf:
The rate of change of magnetic flux (dΦ/dt) gives us the induced emf.
dΦ/dt = ΔΦ / Δt
To find ΔΦ (change in flux) and Δt (change in time), we can subtract the initial values of Φ and t from the final values.

Let's assume the initial time is t = 0 and find the initial flux:
Φ_initial = B_initial * A
Plug in t = 0 into the equation for B:
B_initial = 0.0100(0) + 0.0400(0)^2
B_initial = 0

Φ_initial = B_initial * A
Φ_initial = 0 * 0.003213
Φ_initial = 0 Wb

Now we can calculate the change in flux:
ΔΦ = Φ_final - Φ_initial
ΔΦ = 0.00517 - 0
ΔΦ = 0.00517 Wb

Δt = t_final - t_initial
Δt = 5.40 - 0
Δt = 5.40 s

Finally, we can find the induced emf:
emf = dΦ/dt
emf = ΔΦ / Δt
emf = 0.00517 / 5.40
emf ≈ 0.000956 V (or 0.956 mV)