Two charged particles traverse identical helical paths in a completely opposite sense in a uniform magnetic field $\mathbf{B}=B_0 \hat{\mathbf{k}}$.

Biot-Savart law indicates that the moving electrons (velocity $v$ ) produce a magnetic field $\mathbf{B}$ such that

A current carrying circular loop of radius R is placed in the $x$-y plane with centre at the origin. Half of the loop with $x>0$ is now bent so that it now lies in the $y-z$ plane.

An electron is projected with uniform velocity along the axis of a current carrying long solenoid. Which of the following is true?

In a cyclotron, a charged particle

A circular current loop of magnetic moment $M$ is in an arbitrary orientation in an external magnetic field B. The work done to rotate the loop by $30^{\circ}$ about an axis perpendicular to its plane is

The gyro-magnetic ratio of an electron in an H -atom, according to Bohr model, is

Consider a wire carrying a steady current, $I$ placed in a uniform magnetic field $\mathbf{B}$ perpendicular to its length. Consider the charges inside the wire. It is known that magnetic forces do no work. This implies that,

Two identical current carrying coaxial loops, carry current $I$ in an opposite sense. A simple amperian loop passes through both of them once. Calling the loop as $C$,

A cubical region of space is filled with some uniform electric and magnetic fields. An electron enters the cube across one of its faces with velocity $v$ and a positron enters via opposite face with velocity $-v$. At this instant,

A charged particle would continue to move with a constant velocity in a region wherein,

Verify that the cyclotron frequency $\omega=e B / m$ has the correct dimensions of $[T]^{-1}$.

Show that a force that does no work must be a velocity dependent force.

The magnetic force depends on $v$ which depends on the inertial frame of reference. Does then the magnetic force differ from inertial frame to frame? Is it reasonable that the net acceleration has a different value in different frames of reference?

Describe the motion of a charged particle in a cyclotron if the frequency of the radio frequency (rf) field were doubled.

Two long wires carrying current $I_1$ and $I_2$ are arranged as shown in figure. The one carrying current $I_1$ is along is the $x$-axis. The other carrying current $I_2$ is along a line parallel to the $y$-axis given by $x=0$ and $z=d$. Find the force exerted at $o_2$ because of the wire along the $x$-axis.

A current carrying loop consists of 3 identical quarter circles of radius $R$, lying in the positive quadrants of the $x-y, y-z$ and $z-x$ planes with their centres at the origin, joined together. Find the direction and magnitude of $\boldsymbol{B}$ at the origin.

A charged particle of charge $e$ and mass $m$ is moving in an electric field $\mathbf{E}$ and magnetic field B. Construct dimensionless quantities and quantities of dimension $[T]^{-1}$.

An electron enters with a velocity $\mathbf{v}=v_0 \hat{\mathbf{i}}$ into a cubical region (faces parallel to coordinate planes) in which there are uniform electric and magnetic fields. The orbit of the electron is found to spiral down inside the cube in plane parallel to the $x$-y plane. Suggest a configuration of fields $\mathbf{E}$ and $\mathbf{B}$ that can lead to it.

Do magnetic forces obey Newton's third law. Verify for two current elements $\mathbf{d l}_1=\mathbf{d l} \hat{\mathbf{i}}$ located at the origin and $\mathbf{d l}_2=\mathbf{d l} \hat{\mathbf{j}}$ located at $(0, R, 0)$. Both carry current $I$.

A multirange voltmeter can be constructed by using a galvanometer circuit as shown in figure. We want to construct a voltmeter that can measure $2 \mathrm{~V}, 20 \mathrm{~V}$ and 200 V using a galvanometer of resistance $10 \Omega$ and that produces maximum deflection for current of 1 mA . Find $R_1, R_2$ and $R_3$ that have to be used.

A long straight wire carrying current of 25 A rests on a table as shown in figure. Another wire $P Q$ of length 1 m , mass 2.5 g carries the same current but in the opposite direction. The wire $P Q$ is free to slide up and down. To what height will $P Q$ rise?

A 100 turn rectangular coil $A B C D$ (in $X-Y$ plane) is hung from one arm of a balance figure. A mass 500 g is added to the other arm to balance the weight of the coil. A current 4.9 A passes through the coil and a constant magnetic field of 0.2 T acting inward (in $x-z$ plane) is switched on such that only arm CD of length 1 cm lies in the field. How much additional mass $m$ must be added to regain the balance?

A rectangular conducting loop consists of two wires on two opposite sides of length $l$ joined together by rods of length $d$. The wires are eachof the same material but with cross-sections differing by a factor of 2 . The thicker wire has a resistance $R$ and the rods are of low resistance, which in turn are connected to a constant voltage source $V_0$. The loop is placed in uniform a magnetic field $\mathbf{B}$ at $45^{\circ}$ to its plane. Find $\tau$, the torque exerted by the magnetic field on the loop about an axis through the centres of rods.

An electron and a positron are released from $(0,0,0)$ and $(0,0,1.5 R)$ respectively, in a uniform magnetic field $\mathbf{B}=B_0 \hat{\mathbf{i}}$, each with an equal momentum of magnitude $p=e B R$. Under what conditions on the direction of momentum will the orbits be non-intersecting circles?

A uniform conducting wire of length $12 a$ and resistance $R$ is wound up as a current carrying coil in the shape of (i) an equilateral triangle of side $a$, (ii) a square of sides $a$ and, (iii) a regular hexagon of sides $a$. The coil is connected to a voltage source $V_0$. Find the magnetic moment of the coils in each case.

Consider a circular current-carrying loop of radius $R$ in the $x$-yplane with centre at origin. Consider the line integral

$$\Im(L)=\left|\int_{-L}^L \mathbf{B} \cdot \mathbf{d l}\right|$$

taken along $z$-axis.

(a) Show that $\Im(L)$ monotonically increases with $L$

(b) Use an appropriate amperian loop to show that $\Im(\infty)=\propto_0 I$. where $I$ is the current in the wire

(c) Verify directly the above result

(d) Suppose we replace the circular coil by a square coil of $\operatorname{sides} R$ carrying the same current $I$.

What can you say about $\Im(L)$ and $\Im(\infty)$ ?

A multirange current meter can be constructed by using a galvanometer circuit as shown in figure. We want a current meter that can measure $10 \mathrm{~mA}, 100 \mathrm{~mA}$ and 1 mA using a galvanometer of resistance $10 \Omega$ and that produces maximum deflection for current of 1 mA . Find $S_1, S_2$ and $S_3$ that have to be used.

Five long wires $A, B, C, D$ and $E$, each carrying current $I$ are arranged to form edges of a pentagonal prism as shown in figure. Each carries current out of the plane of paper.

(a) What will be magnetic induction at a point on the axis 0 ? Axis is at a distance $R$ from each wire.

(b) What will be the field if current in one of the wires (say $A$ ) is switched off?

(c) What if current in one of the wire (say $A$ ) is reversed?