g A coil formed by wrapping 50 turns of wire in the shape of a square is positioned in a magnetic field so that the normal to the plane of the coil makes an angle of 30.0° with the direction of the field. When the magnetic field is increased from 250 µT to 700 µT in 0.300 s, an emf of magnitude 60.0 mV is induced in the coil. What is the total length of wire in the coil?

L = 182.4 m

Explanation:

Given:-

- The number of turns of the coil, N = 50

- The shape of the coil = square

- The angle between the coil and magnetic field, θ = 30°

- The change in magnetic field, ΔB = (700 - 250) μT

- The time duration in which magnetic field changes, Δt = 0.3 s

- The induced emf, E = 60.0 mV

Solution:-

- The problem at hand is an application of Faraday's law. The law states that the induced emf (E) is proportional to the negative rate of change of magnetic flux (ΔФ / Δt) and number of turns of the coil (N).

- The Faraday's law is mathematically expressed as:

E = - N * (ΔФ / Δt)

Where,

- The flux (Ф) through a current carrying with an cross-sectional area (A) at a normal angle (θ) to the direction of magnetic field (B) is given by the following relationship.

Ф = B*A*cos (θ)

- We need the rate of change of magnetic flux (ΔФ / Δt) for the Faraday's law. I. e the induced emf (E) is proportional to rate of change in magnetic field (ΔB / Δt), rate of change of angle between the coil and magnetic field (Δθ / Δt) or rate of change of cross-sectional area of the coil under the influence of magnetic field.

- To determine the exact relationship. We will derive the multi-variable function of flux (Ф) with respect to time "t":

Ф (B, A, θ) = B*A*cos (θ)

- The first derivative would be (Use chain and product rules)

(ΔФ / Δt) = ΔB / Δt*A*cos (θ) + B*ΔA/Δt*cos (θ) - B*A*sin (θ) * Δθ/Δt

- For the given problem the only dependent parameter that is changing is magnetic field (B) with respect to time "t". Hence, (ΔA/Δt = Δθ/Δt = 0):

ΔФ / Δt = (ΔB/Δt) * A*cos (θ)

- Substitute the rate of change of magnetic flux (ΔФ / Δt) into the expression for Faraday's Law initially stated:

E = - N * (ΔB/Δt) * A*cos (θ)

- Plug in the values and evaluate the Area of the square coil:

A = - E / (N * (ΔB/Δt) * cos (θ))

A = - 0.06 / (50*[ (250-700) * 10^-6/0.3 ] * cos (30°))

A = - 0.06 / - 0.07216

A = 0.8314 m^2

- The square coil has equal sides (x). The area of a square A is given by:

A = x^2

x = √0.8314

x = 0.912 m

- The perimeter length of a single coil in terms of side length "x" is given as:

P = 4x

Whereas for a coil of N turns the total length (L) would be:

L = N*P

L = 4Nx

L = 4 * 50 * 0.912

L = 182.4 m ... Answer