A magnetic field is perpendicular to the plane of a single-turn circular coil. The magnitude of the field is changing, so that an emf of 0.74 V and a current of 3.8 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 plane of the coil and with a magnitude changing at the same rate). What emf and current are induced in the square coil?
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Changing the loop from a circle to a square will multiply its area by a factor pi/4.
The induced voltage will be multiplied by that factor.
Since the wire resistance will remain unchanged, the induced current will also be multiplied by a pi/4 factor.
To solve this problem, we can use Faraday's Law of electromagnetic induction, which states that the induced emf (ε) in a wire loop is equal to the rate of change of magnetic flux through the loop.
Let's break down the problem step by step:
Step 1: Finding the initial magnetic flux through the circular coil
We are given the emf (ε) induced in the circular coil, which is 0.74 V. Since the magnetic field (B) is perpendicular to the plane of the coil, the magnetic flux (Φ) through the coil is given by the product of the magnetic field and the area of the coil.
Φ = B * A
Since the circular coil has a single turn, the area (A) is simply the area of a circle with radius (r), and A = π * r².
Step 2: Finding the rate of change of magnetic flux
We are not given the exact rate of change of the magnetic field, but we are told that the rate of change of the magnetic flux is the same for both the circular coil and the square coil. Therefore, we can assume that the rate of change of the magnetic field is the same.
Step 3: Finding the magnetic flux through the square coil
After reforming the wire into a single-turn square coil, the area (A) of the coil changes. In a square coil, the area is given by the side length (s), and A = s².
Since the rate of change of the magnetic flux is the same, we can equate the change in flux to find the side length (s) of the square coil.
B * A_circular = B * A_square
B * π * r² = B * s²
Step 4: Finding the emf and current in the square coil
The emf (ε_square) and current (I_square) induced in the square coil can be found using Faraday's Law:
ε_square = -dΦ/dt
I_square = ε_square / R
Where dΦ/dt is the rate of change of the magnetic flux through the square coil, and R is the resistance of the coil.
Since the rate of change of the flux is the same, we can substitute the value of dΦ/dt that we found in Step 3.
ε_square = -(B * s²)/dt
I_square = ε_square / R
Note that the negative sign represents the Lenz's Law, which states that the induced current in a coil will flow in a direction to oppose the change in magnetic flux.
Now you have all the steps needed to find the emf and current induced in the square coil. Plug in the appropriate values into the equations, including the known value of the resistance (if given), and perform the calculations to obtain the final results.