JEE Advanced 2009

LEVELJEE Advanced

Animated Solution for Physics - Electromagnetic Induction: Two metallic rings $A$ and $B$ , identical in shape and size but having different resistivities $ρ_{A}$ and $ρ_{B}$ , are kept on top of two identical solenoids as shown in the figure. When current $I$ is switched on in both the solenoids in identical manner, the rings $A$ and $B$ jump to heights $h_{A}$ and $h_{B}$ , respectively, with $h_{A} > h_{B}$ . The possible relation(s) between their resistivities and their masses $m_{A}$ and $m_{B}$ is (are)

Q: Why is the induced EMF the same for both rings?

The induced EMF depends only on the rate of change of magnetic flux through the area of the ring. Since both solenoids and rings are identical in size and the current is switched on identically, the flux change is exactly the same for both.

Q: Why does the ring jump upwards?

According to Lenz's Law, the induced current in the ring creates a magnetic field that opposes the change in flux. This results in a magnetic polarity opposite to that of the solenoid, causing a strong repulsive force that pushes the ring upwards.

Q: How do we know the impulse is proportional to the induced current?

The magnetic force on the ring is proportional to the product of the induced current and the solenoid's magnetic field. Since the magnetic field change is identical for both setups, the time-integrated force (impulse) is directly proportional to the induced current.

Q: Why is the height inversely proportional to the square of mass and resistivity?

The initial velocity $v$ is proportional to the impulse divided by mass ($v \propto 1/(m\rho)$). From kinematics, the maximum height $h$ is proportional to $v^2$. Squaring the velocity gives $h \propto 1/(m^2\rho^2)$.

Q: Can option (a) be correct if the mass of A is much smaller?

Option (a) explicitly states that $m_A = m_B$. If $\rho_A > \rho_B$ and $m_A = m_B$, then $m_A\rho_A > m_B\rho_B$, which would mean ring A jumps lower than ring B. This contradicts the given condition $h_A > h_B$.

Select Answer:

* Multiple Correct

Visualized Solution

$S e t u p$

$A and B are identical in shape and size.$
$ρ_{A} \neq = ρ_{B} and m_{A}, m_{B} can be different.$

$F a r a d a y^{'} s L a w$

$e = - \frac{d ϕ}{d t}$
$Since dimensions are identical, e_{A} = e_{B} = e$

$I n d u ce d C u r r e n t$

$i = \frac{e}{R}$
$R = ρ \frac{l}{A _{cr oss}} ⟹ R \propto ρ$
$i \propto \frac{1}{ρ}$

$M a g n e t i c I m p u l se$

$F = i \oint d l \times B ⟹ F \propto i$
$J = \int F d t \propto i$

$I ni t ia l V e l oc i t y$

$J = m v ⟹ v = \frac{J}{m}$
$v \propto \frac{i}{m} \propto \frac{1}{m ρ}$

$M a x im u m H e i g h t$

$h = \frac{v ^{2}}{2 g}$
$h \propto \frac{1}{m ^{2} ρ ^{2}}$

$E v a l u a t in g O pt i o n s$

$h_{A} > h_{B} ⟹ \frac{1}{m _{A}^{2} ρ _{A}^{2}} > \frac{1}{m _{B}^{2} ρ _{B}^{2}}$
$m_{A} ρ_{A} < m_{B} ρ_{B}$
$Options (b) and (d) satisfy this condition.$

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The Sigma Insight: Faraday's Laws of Electromagnetic Induction

Solution Diagram

Analyzing the Setup Imagine two identical solenoids, each with a metallic ring resting on top

Ring

A

and Ring

B

are identical in shape and size, but they have different resistivities (

ρ_{A}

and

ρ_{B}

) and possibly different masses (

m_{A}

and

m_{B}

). This classic setup is a beautiful demonstration of the interplay between electromagnetism and classical mechanics.

The Master Equation

Faraday's Law When we switch on the current in the solenoids, a magnetic field suddenly builds up. According to Faraday's Law, this changing magnetic flux induces an electromotive force (EMF) in both rings.

e = - \frac{d ϕ}{d t}

Since the rings have the exact same dimensions and are placed in identical changing magnetic fields, the induced EMF

e

is identical for both. However, their resistivities are different. The induced current

i

is simply the EMF divided by the resistance

R

i = \frac{e}{R}

Since resistance is proportional to resistivity (

R \propto ρ

), the current is inversely proportional to the resistivity (

i \propto \frac{1}{ρ}

The Magnetic Kick By Lenz's Law, this induced current creates a magnetic field that opposes the solenoid's field, resulting in a sudden upward repulsive force

This force acts for a very short time, delivering an upward impulse

J

to the rings.

F = i \oint d l \times B ⟹ F \propto i

J = \int F d t \propto i

This impulse gives the rings an initial upward velocity

v

. Since impulse equals the change in momentum (

J = m v

), the velocity

v

is proportional to the current divided by the mass.

v = \frac{J}{m} \propto \frac{i}{m} \propto \frac{1}{m ρ}

Flight and Final Calculation Now, the rings shoot up into the air like projectiles

The maximum height

h

they reach is given by kinematics as

v^{2} /2 g

. Therefore, the height is inversely proportional to the square of mass times resistivity.

h = \frac{v ^{2}}{2 g} \propto \frac{1}{m ^{2} ρ ^{2}}

The problem states that ring

A

jumps higher than ring

B

(

h_{A} > h_{B}

). This means the product of mass and resistivity for

A

must be strictly less than that for

B

m_{A} ρ_{A} < m_{B} ρ_{B}

Looking at our options: - If

ρ_{A} < ρ_{B}

and

m_{A} = m_{B}

, then

m_{A} ρ_{A} < m_{B} ρ_{B}

holds perfectly. (Option b) - If

ρ_{A} < ρ_{B}

and

m_{A} < m_{B}

, then

m_{A} ρ_{A} < m_{B} ρ_{B}

holds again. (Option d)

Thus, options (b) and (d) are the correct answers.

Common Traps & FAQs

Why is the induced EMF the same for both rings?

The induced EMF depends only on the rate of change of magnetic flux through the area of the ring. Since both solenoids and rings are identical in size and the current is switched on identically, the flux change is exactly the same for both.

Why does the ring jump upwards?

According to Lenz's Law, the induced current in the ring creates a magnetic field that opposes the change in flux. This results in a magnetic polarity opposite to that of the solenoid, causing a strong repulsive force that pushes the ring upwards.

How do we know the impulse is proportional to the induced current?

The magnetic force on the ring is proportional to the product of the induced current and the solenoid's magnetic field. Since the magnetic field change is identical for both setups, the time-integrated force (impulse) is directly proportional to the induced current.

Why is the height inversely proportional to the square of mass and resistivity?

The initial velocity