Option 4 : 200

__Concept:__

According to Faraday's law of electromagnetic induction, the induced emf is equal to the rate of change of flux multiplied by the number of turns.

\(E = -N\frac{dϕ}{dt}\)

The negative sign is due to Lenz's law.

__Calculation:__

Given voltage is 50 V, dϕ = 70 - 20 = 50 mWb, dt = 0.2 s

Total number of turns is:

\(N = \frac{|E|}{|d\phi|}|dt|\)

\(N=\frac{50}{50 \times 10^{-3}}0.2\)

**N = 200**

Option 1 : 6.28 × 10^{-1}T

__CONCEPT:__

- Solenoid: A type of electromagnet that generates a controlled magnetic field through a coil wound into a tightly packed helix.
- The uniform magnetic field is produced when an electric current is passed through it.

- The
**magnetic field inside a solenoid is proportional to the applied current and the number of turns per unit length**. - The magnetic field inside a solenoid does not depend on the diameter of the solenoid.
**The field inside is constant.**

B = μ_{0}N I

where N is no. of turns per unit length, I is current in the solenoid and μ_{0} is the permeability of free space.

**EXPLANATION:**

Given that I = 5 A

length of solenoid = 0.4 cm = 0.004 m

no. of turns per unit length N = 400 / 0.004 = 10^{5}

constant μ_{0} = 4π × 10^{-7}

** **Mgnetic field inside = μ_{0} N I = 4π × 10-7 × 10^{5 }× 5

The magnetic field inside the solenoid = **6.28 × 10-1 T**

- The magnetic field outside a solenoid is zero.

Option 4 : 200

__Concept:__

According to Faraday's law of electromagnetic induction, the induced emf is equal to the rate of change of flux multiplied by the number of turns.

\(E = -N\frac{dϕ}{dt}\)

The negative sign is due to Lenz's law.

__Calculation:__

Given voltage is 50 V, dϕ = 70 - 20 = 50 mWb, dt = 0.2 s

Total number of turns is:

\(N = \frac{|E|}{|d\phi|}|dt|\)

\(N=\frac{50}{50 \times 10^{-3}}0.2\)

**N = 200**

Two solenoids have identical geometrical construction but one is made of thick wire and the other of thin wire. Which of the following quantities are different for the two solenoids?

i. self-inductance

ii. rate of Joule heating if the same current goes through them

iii. magnetic field energy if the same current goes through them

iv. time constant

Option 4 : ii & iv

__Solenoid:__

- Solenoid is a type of electromagnet which is a coils of wire.

- Its purpose is to generate a magnetic field which is controlled by some value of current through a coil which is wound into a tightly packed helix.
- The coil can be arranged to produce a uniform magnetic field in a vacuum of space when an electric current is passed through it.

In the question given that there are two solenoids.

- One is made of thick wire

- Other is made of thin wire

⇒ Area of thick wire (A_{1}) > area of thin wire (A_{2})

Now parameters for comparison of the solenoids are

**i) Self inductance (L):**

- \(L = \frac{\phi }{I} = \frac{{\mu {N^2}\left( {Hollow\;area\;of\;solenoids} \right)}}{{length\;of\;wire}}\)
- From given formula of self inductance, it is clear that it does not depends on the thickness of wire of solenoids, hence it is remain same.

**ii) Rate of Joule heating if the same current goes through them:**

Joule heating, **H = i ^{2} Rt**

And we know that, \(R = \rho \frac{l}{A} \Rightarrow R \propto \frac{1}{A}\)

Since, **A _{1} > A_{2}, then R_{1} < R_{2} **

⇒ **Joule heating of two solenoids is different for same current.**

**iii) Magnetic field energy if the same current goes through them:**

Magnetic field energy - \(\left( M \right) = \frac{1}{2}L{i^2}\)

⇒ M_{1} = M_{2}

**As the magnetic field does not depends on the thickness of the wire. Hence this energy is same for both wire.**

**iv) Time constant (τ):**

\(\tau = \frac{L}{R} \Rightarrow \tau \propto \frac{1}{R}\;and\;R \propto \frac{1}{A}\)

A_{1} > A_{2} ⇒ R_{1} < R_{2} and hence τ_{1} > τ_{2}

**Therefore time constant of both the solenoid is different as it depends on the resistance of solenoid which is ultimately depends on the area of wire.**