Bonus problem – JEE First

Biot-Savart law on a polygon

September 30th, 2022jeefirst0 Comments

A current $I$ flows along a thin wire, shaped as a regular polygon with $N$ sides, which can be inscribed intro a circle of radius $R$ . Find the magnetic field at the center of the polygon. What happens if $N$ is made very large?

Solution:

The polygon described in the problem is illustrated in the figure below, for $N=8$ . The magnetic field created by this structure is the superposition of fields from $N$ straight current carrying wires of length $2 R \sin (\pi/N)$ . Therefore, we will need to find the magnetic field due to a wire of finite length first.

Rendered by QuickLaTeX.com

The Biot-Savart law gives the magnetic field …

Buoyancy of connected objects

July 16th, 2022jeefirst0 Comments

JEE Advanced 2013 Paper 1, Question 12

A solid sphere of radius $R$ and density $\rho$ is attached to one end of a mass-less spring of force constant $k$ . The other end of the spring is connected to another solid sphere of radius $R$ and density $3 \rho$ . The complete arrangement is placed in a liquid of density $2 \rho$ and is allowed to reach equilibrium. The correct statement(s) is (are)

the net elongation of the spring is $\frac{4 \pi R^{3} \rho g}{3 k}$ .
the net elongation of the spring is $\frac{8 \pi R^{3} \rho g}{3 k}$
the light sphere is partially submerged.
the light sphere is completely submerged.

Solution

Recall that the buoyant force experienced by …

Electromagnetic induction in a twisted loop

February 21st, 2022jeefirst0 Comments

JEE Advanced 2017 Paper 1, Question 5

A circular insulated copper wire loop is twisted to form two loops of area $A$ and $2 A$ as shown in the figure. At the point of crossing the wires remain electrically insulated from each other. The entire loop lies in the plane (of the paper). A uniform magnetic field ${\bf B}$ points into the plane of the paper. At $t=0$ , the loop starts rotating about the common diameter as axis with a constant angular velocity $\omega$ in the magnetic field. Which of the following options is/are correct?

Rendered by QuickLaTeX.com

The emf induced in the loop is proportional to the sum

…

Mass on a semicircular block

February 1st, 2022jeefirst0 Comments

A heavy particle of mass $m$ is placed at the top of a semicircular block of radius $R$ . Find the height at which the particle falls off, assuming (i) the block is fixed to the ground, and (ii) the block has a mass $M$ and is free to move. Assume all surfaces are frictionless.

Rendered by QuickLaTeX.com

Related problem: Sliding on a block with a circular cut.

Solution:

(i) We first consider the case where the block is fixed to the ground. As the mass slides down the block, there are three forces acting on it: the weight $mg$ , the centrifugal force $m R \dot{\theta}^2$ , and …

A spherical capacitor

January 20th, 2022jeefirst0 Comments

A spherical conducting shell with radius $b$ is concentric with a conducting ball with radius $a$ , with $a<b$ .

Compute the capacitance $C = Q / \Delta \phi$ when the shell is grounded and the ball has charge $Q$ .
Compute the capacitance when the ball is grounded and the shell has charge $Q$ .
Compute the full matrix of coefficients of capacitance for the two conductors.
Considering these conductors as a capacitor, determine its capacitance. That is, assign equal and opposite charges $\pm Q$ to the shell and the ball, and compute $C = Q / \Delta \phi$ .

Related Problem: Insulating spherical shell with a hole

Solution

(a) First, we ground the shell and give the …