Please help, show all the works for all the questions, please. [B] A toroid is w

Question

Please help, show all the works for all the questions, please.

[B] A toroid is wound on a ferrite core with a square cross section. The ring is 5mm thick and has an inner radius of 5mm. The toroid is wound with 20 turns. The ferrite has a relative permeability of 104 which multiplies Uo in the calculations. Calculate the magnetic field in terms of the current I. Set up the calculation using Ampere's Law. Calculate the magnetic flux in terms of the current I. I Determine the inductance of the toroid. The toroid is connected in series with a 100kOhm resistor, a switch and a 9V battery. Write the KirchofF s law expression for this circuit and calculate the current after the switch has been closed for a long time.

Explanation / Answer

a) we have by ampere's law

l = thickness = 5 mm = 0.005 m

N - number of loop = 20

B* (l) = (10^4) *u0*I * N

so, B * (0.005) = (10^4) *4 * 3.14 *(10^-7)*20 * I

B = 50.24 * I

magnetic filed strength , B ( in tesla) = 50.24 * I ( in amperes)

B = 50.24 * I

b)

magnetic flux = B * area = 50.24 * I* (pi *r^2) = 50.24 *3.14 * (0.005^2) *I = 0.0039438* I

magnetic flux (in webers) = 0.0039438* I

c)

we have inductance = (flux * N)/(current) = (0.0039438* I * 20)/I = 0.0788768 Henry

d)

Kirchoff's expression
V = IR + ( L * dI/dt)
V = 9
R = 100*10^3 ohms
L = 0.0788768 H
so, 9 = ( i * 10^5) + ( 0.0788768 * di/dt)
I = (9/10^5) * ( 1- e^ ( -t * (10^5 / 0.0788768)))
I = (9/10^5) * ( 1- e^ ( -1267799.91*t))

for a long time t = infinty

so I = (9/10^5) = 9 *10^-5 Amperes is the current after a long time

e)

expression of current as a function of time

I = (9/10^5) * ( 1- e^ ( -1267799.91*t))

for half time

I = 0.5 * (9/10^5)

so 0.5 * (9/10^5) = (9/10^5) * ( 1- e^ ( -1267799.91*t))

0.5 =  ( 1- e^ ( -1267799.91*t))

t = 5.467 *10^-7 seconds

half time = t = 5.467 *10^-7 seconds is the answer

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