You have a fixed length of conducting wire. Using it you can construct a single

Question

You have a fixed length of conducting wire. Using it you can construct a single turn flat coil that has the shape of a square, a circle, or a rectangle with the long side twice the length of the short side. Each can be used with the same magnetic field to produce a generator that operates at the same frequency. Rank the peak emfs epsilon_0 of the three generators in descending order (largest first). epsilon_0, square, epsilon_0, circle, epsilon_0, rectangle epsilon_0, square epsilon_0, rectangle, epsilon_0, circle epsilon_0, rectangle, epsilon_0, square, epsilon_0, circle epsilon_0, circle, epsilon_0, square, epsilon_0, rectangle epsilon_0, rectangle, epsilon_0, circle, epsilon_0, square

Explanation / Answer

4) Eo,rectangle > Eo,cirlce > Eo,square

let L is the length of the conducting wire.

let r is the radius of the cirlce.

so, L = 2*pi*r

r = L/(2*pi)

Area of the circular loop, A_circle = pi*r^2

= pi*(L/(2*pi))^2

= L^2/(4*pi)

let a is length of each side of the square.

4*a = L

a = L/4

area of the square loop,

A_square = a^2 = L^2/16

let a is the length of the short of rectangle

b = 2*a (another side)

so, 2*a + 2*b = L

2*a + 2*2*a = L

6*a = L

a = L/6

Area of the rectangle, A_rectangle = a*b

= a*2*a

= 4*a^2

= 4*(L/6)^2

= L^2/9

clearly, A_rectangle > A_cirlce > A_square

we know, maximum induced emf in a coil, emf_max = N*A*B*w

so,

Eo,rectangle > Eo,cirlce > Eo,square

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