Question

A toroid is a long coil of wire wound over a circular core. If 'r' and 'R' are the radii of the coil and toroid respectively, the coefficient of self-induction of the toroid is (The magnetic field in it is uniform and R > > r) ____________.

(N = number of turns of the coil and µ₀ = permeability of free space)

Options

• `(mu_0  "R")/(2  "N"^2 "r"^2)`

• `(mu_0  "N"^2 "R"^2)/(2 "r")`

• `(mu_0  "N"^2 "r"^2)/(2 "R")`

• `(2mu_0  "r"^2)/("N"^2 "R")`

Answer 1

A toroid is a long coil of wire wound over a circular core. If 'r' and 'R' are the radii of the coil and toroid respectively, the coefficient of self-induction of the toroid is `(mu_0  "N"^2 "r"^2)/(2 "R")`.

Explanation:

`"L" = phi/"I"`

`phi = "NAB"`

`"B" = mu_0 "nI"`

`"n" = "N"/(2 pi"R")`

` therefore phi = "N"pir^2 (mu_0 "N"/(2pi"R") "I")`

`= (mu_0 "N"^2 "r"^2 "I")/(2"R")`

`"L" = phi/"I" = (mu_0 "N"^2 "r"^2)/(2"R")`