The magnetic field shown in the figure decreases from 1.0 T to 0.4 T in 1.2 s. A
6.0 cm diameter loop with a resistance of 0.010 W is perpendicular to the field.
a. Find the average emf induced in the coil during this time.
b. What is the size and direction of the current induced in the loop?
To find the average electromotive force (emf) induced in the coil, you can use Faraday's law of electromagnetic induction, which states that the emf induced in a circuit is equal to the rate of change of magnetic flux through the circuit.
a. First, we need to determine the change in magnetic flux (\Delta\Phi). The formula to calculate magnetic flux (\Phi) is given by:
\Phi = B * A * cos(\theta),
where B is the magnetic field, A is the area of the loop, and \theta is the angle between the magnetic field direction and the normal vector to the loop.
In this case, the magnetic field decreases from 1.0 T to 0.4 T, so the change in magnetic field (\Delta B) is 0.4 T - 1.0 T = -0.6 T.
The area of the loop (A) can be calculated using the diameter (d) of the loop:
A = \pi * (d/2)^2
Given that the diameter is 6.0 cm, we can convert it to meters:
d = 6.0 cm = 0.06 m
Now, we can calculate the area:
A = \pi * (0.06/2)^2 = 0.00282743338 m^2
Since the field is perpendicular to the loop (\theta = 90 degrees), the cosine term is 1.
\Delta\Phi = \Delta B * A
= -0.6 T * 0.00282743338 m^2
= -0.00169646 Wb (webers)
The negative sign signifies a decrease in flux.
Now we can calculate the average emf using the time (t) it takes for the magnetic field to change. The average emf (\varepsilon) is given by:
\varepsilon = \frac{\Delta\Phi}{\Delta t},
where \Delta t = 1.2 s.
\varepsilon = \frac{-0.00169646 Wb}{1.2 s}
= -0.001413717 Wb/s (webers per second)
The negative sign indicates the direction of the induced current, which we will find in the next part.
b. To find the size and direction of the induced current, we can use Ohm's Law, which states that the current (I) flowing through a circuit is equal to the emf (E) divided by the resistance (R):
I = \frac{\varepsilon}{R},
where R = 0.010 W (ohms).
Note that the emf (\varepsilon) we calculated in part a is the average emf.
I = \frac{-0.001413717 Wb/s}{0.010 W}
= -0.1413717 A (amperes)
Again, the negative sign indicates the direction of the current. In this case, the induced current flows in the opposite direction to maintain the magnetic field.
Therefore, the average emf induced in the coil during this time is approximately -0.001413717 Wb/s, and the size and direction of the current induced in the loop is approximately -0.1413717 A (opposite the direction of the original magnetic field).