Q. 655.0( 3 Votes )

# The current in an ideal, long solenoid is varied at a uniform rate of 0.01 A s^{–1}. The solenoid has 2000 turns/m and its radius is 6.0 cm.

(a) Consider a circle of radius 1.0 cm inside the solenoid with its axis coinciding with the axis of the solenoid. Write the change in the magnetic flux through this circle in 2.0 seconds.

(b) Find the electric field induced at a point on the circumference of the circle.

(c) Find the electric field induced at a point outside the solenoid at a distance 8.0 cm from its axis.

Answer :

Given:

Rate of variation of current = 0.01 A s^{–1}.

No of turns/m (n) = 2000

Radius(r) = 6 cm = 0.06 m

Formula used:

(a) Radius of circle(r’) = 1 cm = 0.01 m

Time(t) = 2 s

For two seconds, change of current = (2 x 0.01 A.) = 0.02 A

Magnetic flux , where B = magnetic field, A = area

Area of circle

Magnetic field of a solenoid , where μ_{0} = magnetic permeability of vacuum, n = number of turns per unit length, Δi = change in current

Hence, flux =

⇒ Wb

Hence, in 1 second = **Wb** (Ans)

(b) = , where E = electric field, dr = line element, E’ = emf, = flux, t = time

Hence, in this case, this becomes

, where r = radius of circle

⇒ **Vm ^{-1}**(Ans)

(c) For the point located outside,

Wbs^{-1}

= flux, t = time, μ_{0} = magnetic permeability of vacuum, n = number of turns per unit length, di/dt = rate of change in current

⇒ , where E = electric field, r = radius of circle(since )

Hence,

⇒ Vm^{-1} (Ans)

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The current in a solenoid of 240 turns, having a length of 12 cm and a radius of 2 cm, changes at a rate of 0.8 A s^{–1}. Find the emf induced in it.

Figure shows a square frame of wire having a total resistance r placed co-planarly with long, straight wire. The wire carries a current I given by i = i_{0} sin ωt. Find

(a) the flux of the magnetic field through the square frame,

(b) the emf induced in the frame and

(c) the heat developed in the frame in the time interval 0 to _{}.

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