A generator uses a coil that has 280 turns and a 0.45-T magnetic field. The frequency of this generator is 60.0 Hz, and its emf has an rms value of 120 V. Assuming that each turn of the coil is a square (an approximation), determine the length of the wire from which the coil is made.
Use the induced E.M.F. to get the product N*A, where A is the coil area and N is the number of turns (280).
Erms = N*w*A*B/sqrt2
w = coil angular frequency = 377 rad/s
The 1/sqrt2 factor relates the rms voltage to the amplitude.
B = 0.45T
You already know N so that gives you the value of A. The length of a side of the square loop is sqrtA.
4*N*sqrtA is the length of wire.
2(pi)(frequency) = w
To determine the length of the wire from which the coil is made, we can use the formula:
emf = N * ∆Φ/∆t
Where:
emf is the rms value of the electromotive force (in volts),
N is the number of turns in the coil,
∆Φ is the change in magnetic flux through the coil, and
∆t is the time interval over which the change occurs.
In this case, the magnetic field (B) is constant, so the change in magnetic flux (∆Φ) is given by:
∆Φ = B * A
Where:
B is the magnetic field strength (in Tesla)
A is the area of the coil (in square meters).
Since each turn of the coil is square, the area of one turn is equal to the side length squared (s²), so:
A = s²
Substituting these equations into the original formula, we have:
emf = N * B * A/∆t
120 V = 280 * 0.45 T * s²/∆t
We need to solve for the side length (s). To do this, we'll rearrange the equation:
s² = (120 V * ∆t)/(280 * 0.45 T)
Now, we know that the frequency (f) is given by:
f = 1/∆t
So, we can rewrite the equation as:
s² = (120 V * 1/f)/(280 * 0.45 T)
Substituting the given frequency of 60.0 Hz:
s² = (120 V * 1/60.0 s)/(280 * 0.45 T)
Now, we can solve for s by taking the square root of both sides of the equation:
s = √((120 V * 1/60.0 s)/(280 * 0.45 T))
Evaluating this equation will give us the side length of the square turn. To determine the length of the wire from which the coil is made, we need to multiply the side length by the number of turns:
Length of wire = s * N
Substituting the values of s and N will give us the final answer.