A five turn square loop (10 cm on an edge and resistance = 4.0 ohms) is placed in a magnetic field that makes an angle of 30 degrees with the plane of the loop. The magnitude of this field varies with the time according to B = 0.50 t^2, where t is measured in seconds and B in Tesla, T. What is the induced current in the coil at t = 4.0 s?
Well, aren't you buzzing with electrifying questions today! Let's juggle some numbers and solve this magnetic mystery, shall we?
The induced current in a coil is given by the equation I = -N * A * dB/dt, where I is the induced current, N is the number of turns in the coil, A is the area of the loop, and dB/dt is the rate of change of magnetic field.
First, let's find the rate of change of magnetic field at t = 4.0 s. We're given that B = 0.50 * t^2, so at t = 4.0 s:
B = 0.50 * (4.0)^2
B = 0.50 * 16.0
B = 8.0 T
Now, let's calculate the area of the loop. The loop is square, with each side measuring 10 cm or 0.10 m. So, the area of the loop is:
A = (0.10)^2
A = 0.01 m^2
The number of turns in the coil is given as 5.
Finally, let's calculate the induced current using our magic equation:
I = -N * A * dB/dt
I = -5 * 0.01 * (dB/dt)
Substituting the value of dB/dt we found earlier:
I = -5 * 0.01 * (8.0)
I = -0.40 A
So, the induced current in the coil at t = 4.0 s is -0.40 Amperes. Ah, electricity never fails to amuse us with its shocking antics!
To find the induced current in the coil at t = 4.0 s, we need to follow these steps:
Step 1: Calculate the magnetic flux through the loop at t = 4.0 s.
Step 2: Use Faraday's law of electromagnetic induction to find the induced electromotive force (emf) in the loop.
Step 3: Use Ohm's law to find the induced current in the coil.
Let's go through these steps one by one.
Step 1: Calculate the magnetic flux through the loop at t = 4.0 s.
The magnetic flux (Φ) through a loop is given by the formula:
Φ = B * A * cos(θ),
where B is the magnetic field, A is the cross-sectional area of the loop, and θ is the angle between the magnetic field and the plane of the loop.
Given:
B = 0.50 t^2 (in Tesla),
A = (10 cm)^2 = 100 cm^2 = 0.01 m^2 (cross-sectional area of the loop), and
θ = 30 degrees.
Plugging in the values:
Φ = (0.50 * (4.0 s)^2) * (0.01 m^2) * cos(30 degrees).
Simplifying the equation:
Φ = (0.50 * 16.0) * (0.01) * (√3 / 2).
Φ = 0.4 * 0.01 * (√3 / 2).
Φ = 0.004 * (√3 / 2).
Step 2: Use Faraday's law of electromagnetic induction to find the induced emf in the loop.
Faraday's law states:
emf = -d(Φ)/dt,
where emf is the induced electromotive force and d(Φ)/dt is the rate of change of magnetic flux.
Taking the derivative of Φ with respect to time:
d(Φ)/dt = 0.004 * (√3 / 2) * (-0.50) * 2t.
Plugging in the value t = 4.0 s:
d(Φ)/dt = 0.004 * (√3 / 2) * (-0.50) * 2(4.0 s).
d(Φ)/dt = -0.004 * (√3 / 2) * (-2) * (4.0 s).
d(Φ)/dt = 0.004 * (√3 / 2) * 4.0 s.
Step 3: Use Ohm's law to find the induced current in the coil.
Ohm's law states:
emf = I * R,
where emf is the induced electromotive force, I is the induced current in the coil, and R is the resistance of the coil.
We are given the resistance R = 4.0 ohms.
Plugging in the values:
0.004 * (√3 / 2) * 4.0 s = I * 4.0 ohms.
Simplifying the equation:
0.004 * (√3 / 2) * s = I.
Finally, to find the induced current at t = 4.0 s, plug in t = 4.0 s:
I = 0.004 * (√3 / 2) * 4.0 s.
Calculating the value:
I = 0.004 * (√3 / 2) * 4.0.
I ≈ 0.0139 A.
Therefore, the induced current in the coil at t = 4.0 s is approximately 0.0139 Amperes.
To find the induced current in the coil at t = 4.0 s, we can use Faraday's law of electromagnetic induction. This law states that the induced electromotive force (EMF) in a closed loop is equal to the rate of change of the magnetic flux through the loop.
The magnetic flux through the loop can be calculated using the equation:
Φ = B * A * cos(θ)
Where:
- Φ is the magnetic flux
- 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
In this case, the magnetic field (B) varies with time according to B = 0.50 * t^2, where t is measured in seconds, and the area of the loop (A) is given by A = (10 cm)^2 = 100 cm^2 = 0.01 m^2.
We need to determine the magnetic flux at t = 4.0 s, so we substitute the given values:
B = 0.50 * (4.0)^2 = 0.50 * 16 = 8.0 T (Tesla)
A = 0.01 m^2
Now, we need to determine the angle (θ) between the magnetic field and the plane of the loop. In this case, it is given that the angle is 30 degrees.
θ = 30 degrees = (30 * π) / 180 = 0.52 radians
Now, we can calculate the magnetic flux:
Φ = B * A * cos(θ)
= 8.0 * 0.01 * cos(0.52)
≈ 0.079 T * m^2
Finally, the induced electromotive force (EMF) can be calculated using the rate of change of magnetic flux with time:
EMF = dΦ / dt
To find dΦ/dt, we differentiate the magnetic flux equation with respect to time:
dΦ/dt = dB/dt * A * cos(θ)
Since given B = 0.50 * t^2, we can differentiate it:
dB/dt = d(0.50 * t^2) / dt
= 1.0 * t
= t
Now, we substitute the values of dB/dt, A, and cos(θ) into the equation:
EMF = dΦ / dt
= t * 0.01 * cos(0.52)
≈ 0.01t * cos(0.52) T * m^2/s
Finally, to find the induced current, we can use Ohm's law, which relates the induced EMF to the current and resistance:
EMF = I * R
Solving for I, we have:
I = EMF / R
I = (0.01t * cos(0.52)) / 4.0
Now, substituting t = 4.0 s into the equation, we can calculate the induced current:
I = (0.01 * 4.0 * cos(0.52)) / 4.0
≈ 0.01 * 4.0 * 0.862 / 4.0
≈ 0.0086 A
Therefore, the induced current in the coil at t = 4.0 s is approximately 0.0086 Amperes.