A circular coil of 30 turns and radius 7 cm is placed with its plane oriented at 90 degrees to a magnetic field of 0.1 T. the fieldis now increased at a steady rate, reaching a value of 0.7 T after 5 seconds. when emf is induced in the coil
V = Area * (Number of turns)*(Rate of change of B-field)
= 30*pi*(.07m)^2*(0.6 T)/5s
= ___ Volts
To find out when the emf is induced in the coil, we can use Faraday's Law of electromagnetic induction. According to Faraday's Law, the induced emf in a coil is directly proportional to the rate of change of magnetic flux through the coil.
The magnetic flux through a coil can be calculated using the equation:
Φ = B * A * cos(theta)
Where:
Φ is the magnetic flux
B is the magnetic field strength
A is the area of the coil
theta is the angle between the magnetic field and the normal to the coil's surface.
In this case, the magnetic field is increasing at a steady rate from 0.1 T to 0.7 T over 5 seconds. The area of the coil is given by the formula A = π*r^2, where r is the radius of the coil. The angle theta is 90 degrees.
Let's calculate the magnetic flux at different times to find out when the emf is induced.
1. Initial magnetic flux (t = 0 seconds):
Using the given values:
B = 0.1 T
A = π * (0.07 m)^2
theta = 90 degrees = π/2 radians
Calculating the magnetic flux:
Φ = 0.1 * π * (0.07)^2 * cos(π/2)
Φ = 0.0966 Wb
2. Final magnetic flux (t = 5 seconds):
Using the given values:
B = 0.7 T
A = π * (0.07 m)^2
theta = 90 degrees = π/2 radians
Calculating the magnetic flux:
Φ = 0.7 * π * (0.07)^2 * cos(π/2)
Φ = 0.67703 Wb
The emf induced in the coil occurs when there is a change in magnetic flux. So, the emf is induced between the initial and final magnetic flux values, i.e., between 0.0966 Wb and 0.67703 Wb.
Therefore, the emf is induced in the coil during the 5-second period while the magnetic field is increasing.
To determine when the electromotive force (emf) is induced in the coil, we need to consider Faraday's law of electromagnetic induction. According to this law, the emf induced in a coil is equal to the rate of change of magnetic flux through the coil.
The magnetic flux (Φ) through a coil with turns (N) and area (A) is given by the formula: Φ = B * A * cos(θ), where B is the magnetic field strength, A is the area of the coil, and θ is the angle between the coil's plane and the magnetic field.
In this case, we are given that the coil has 30 turns (N = 30) and a radius of 7 cm. The area of the coil (A) can be calculated using the formula for the area of a circle: A = π * r^2, where r is the radius of the coil.
First, let's calculate the area of the coil:
r = 7 cm = 0.07 m (converting from centimeters to meters)
A = π * (0.07)^2
A ≈ 0.0154 m^2
Given that the plane of the coil is oriented at 90 degrees (perpendicular) to the magnetic field, θ = 90 degrees.
Now, let's determine the initial magnetic flux (Φ_1) when the magnetic field is 0.1 T:
Φ_1 = B_1 * A * cos(θ)
Φ_1 = 0.1 T * 0.0154 m^2 * cos(90 degrees)
Φ_1 = 0.1 T * 0.0154 m^2 * 0
Φ_1 = 0
Since the initial magnetic flux is zero, there is no emf induced in the coil at the beginning.
Next, we need to determine the final magnetic flux (Φ_2) when the magnetic field increases to 0.7 T after 5 seconds:
Φ_2 = B_2 * A * cos(θ)
Φ_2 = 0.7 T * 0.0154 m^2 * cos(90 degrees)
Φ_2 = 0.7 T * 0.0154 m^2 * 0
Φ_2 = 0
Similar to the initial magnetic flux, the final magnetic flux is also zero.
Therefore, the emf is induced in the coil whenever there is a change in magnetic flux through the coil. It does not depend on specific time intervals but rather on the change in magnetic field strength or orientation relative to the coil. In this case, since there is no change in magnetic flux, there is no emf induced in the coil.