A magnetic field is perpendicular to the plane of a single turn circular coil. The magnitude of the field is changing so that the emf of 0.80 V and a current of 3.2 A are induced in the coil. The wire is then re-formed into a single turn square coil, which is used in the same magnetic field (again perpendicular to the place of the coil and with the magnitude changing at the same rate). What emf and current are induced in the square coil?
To find the emf and current induced in the square coil, we can use Faraday's Law of electromagnetic induction. According to Faraday's Law, the emf induced in a coil is directly proportional to the rate of change of magnetic flux through the coil.
1. First, let's calculate the initial magnetic flux through the circular coil. The magnetic flux is given by the product of the magnetic field strength perpendicular to the coil and the area of the coil.
Magnetic flux (Φ) = magnetic field strength (B) × area (A)
Given that the emf induced in the circular coil is 0.80 V, we have:
emf (ε) = - N d(Φ)/dt [From Faraday's Law, where N is the number of turns in the coil]
Rearranging the equation, we have:
-ε = N d(Φ)/dt
-0.80 V = 1 turn × d(Φ)/dt
Since the number of turns for the circular coil is 1, we can simplify the equation to:
-0.80 V = d(Φ)/dt
2. The same magnetic field is applied to the square coil, and the rate of change of magnetic flux will remain the same. Therefore, the emf induced in the square coil will be the same as in the circular coil, which is -0.80 V.
emf (ε) = -0.80 V
3. To find the current induced in the square coil, we can use Ohm's Law. Ohm's Law states that the current (I) through a conductor is equal to the emf (ε) divided by the resistance (R).
I = ε / R
The resistance of the coil depends on its dimensions and the resistivity of the material. Since the wire is re-formed into a square coil, the area of the coil will change, but the resistivity of the wire and the length of the wire remains the same. The resistance of the square coil will be different from the circular coil.
4. Since we don't have information about the dimensions of the square coil or the resistivity of the wire, we cannot determine the exact resistance. Therefore, we cannot calculate the current induced in the square coil.
In conclusion, the emf induced in the square coil will be -0.80 V, similar to the circular coil. However, we cannot determine the current induced in the square coil without additional information about its dimensions and the resistivity of the wire.
To find the emf and current induced in the square coil, we can use Faraday's Law of Electromagnetic Induction. According to Faraday's Law, the induced emf in a circuit is directly proportional to the rate at which the magnetic field through the circuit changes.
Let's break down the problem step by step:
1. For the single turn circular coil, the given emf is 0.80 V and the current is 3.2 A. We can assume that the rate of change of the magnetic field is the same for both coils.
2. Faraday's Law states that the induced emf (ε) is equal to the negative rate of change of magnetic flux (Φ) through the circuit. Mathematically, ε = -dΦ/dt.
3. For a single turn coil, the magnetic flux through the coil (Φ) is given by the equation Φ = B * A, where B is the magnetic field and A is the area of the coil.
4. Since the coil is single turn, the area (A) of the coil is equal to the area of the circle formed by the coil.
5. Now, let's consider the square coil. The area (A') of the square coil is given by the equation A' = s^2, where s is the side length of the square coil.
6. With the same magnetic field and rate of change as the circular coil, we can write the equation ε' = -dΦ'/dt for the square coil.
7. The magnetic flux (Φ') through the square coil is Φ' = B * A' = B * s^2, where B is the magnetic field.
8. Since the rate of change of the magnetic field is the same, we can write dΦ/dt = dΦ'/dt.
9. Solving the equations ε = -dΦ/dt and Φ' = B * s^2, we can find the emf and current induced in the square coil.
ε' = -dΦ'/dt = -d(B*s^2)/dt = -2Bs * (ds/dt)
Thus, the emf induced in the square coil is given by ε' = -2Bs * (ds/dt).
To find the current induced in the square coil, we can use Ohm's Law, which states that current (I) is equal to emf (ε) divided by the resistance (R) in the circuit. In this case, R can be assumed to be constant.
I' = ε' / R.
By substituting the value of ε' from the previous equation, we can find the current induced in the square coil.
I' = (-2Bs * (ds/dt)) / R.
Note: The negative sign indicates that the induced current will flow in the opposite direction of the changing magnetic field, according to Lenz's Law.
By evaluating these equations with the given values for the magnetic field, emf, and current of the circular coil, we can find the emf and current induced in the square coil.