Question

(a) State Lenz’s law. Give one example to illustrate this law. “The Lenz’s law is a consequence of the principle of conservation of energy.” Justify this statement.
(b) Deduce an expression for the mutual inductance of two long coaxial solenoids but having different radii and different number of turns.

Solution

(a) According to Lenz’s law, the direction of the induced current (caused by induced emf) is always such as to oppose the change causing it.

ε = - k \frac{dϕ}{dt}

where k is a positive constant. The negative sign expresses Lenz’s law. It means that the induced emf is such that, the induced current opposes the change in flux, if the circuit is closed.
Consider a bar magnet and a loop. The bar magnet experiences a repulsive force due to the current induced. Hence, some amount of work is done to move the magnet. The energy which is spent by the person in moving the magnet is dissipated by Joul's heating produced by induced current. Therefore, the law of conservation of energy is validated.

(b) Figure shows a coil of N₂ turns and radius R₂ surrounding a long solenoid of length l₁, radius R_l and number of turns N₁.

To calculate mutual inducatnce between them, let us assume a current i₁through the inner solenoid S₁There is no magnetic field outside the solenoid and the field inside has magnitude,

B = μ_{0} (\frac{N_{1}}{l_{1}}) i_{1}

and is directed parallel to the solenoid’s axis.

The magnetic flux Φ_B2 through the surrounding coil is, therefore,

ϕ_{B 2} = B ({πR}_{1}^{2}) = \frac{μ_{0} N_{1} i_{1}}{l_{1}} {πR}_{1}^{2}

Now,

M = \frac{N_{2} ϕ_{B 2}}{i_{1}} = (\frac{N_{2}}{i_{1}}) (\frac{μ_{0} N_{1} i_{1}}{l_{1}}) {πR}_{1}^{2} = \frac{μ_{0} N_{1} N_{2} {πR}_{1}^{2}}{l_{1}} M = \frac{μ_{0} N_{1} N_{2} {πR}_{1}^{2}}{l_{1}}

We can see from the above deduced experssion that, mutual inducatnce is independent of the radius R₂ of the surrounding coil. This is because, solenoid's magnetic field is confined only to its interior side.

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Electric Charges And Fields

Some More Questions From Electric Charges and Fields Chapter

a) An electrostatic field line is a continuous curve. That is, a field line cannot have sudden breaks. Why not?
b) Explain why two field lines never cross each other at any point?

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