1.
A proton and an alpha particle enter a uniform magnetic field perpendicularly with the same momentum. Deduce the ratio of the radii of their respective circular paths.
2.
An electron is moving with a velocity $\vec{v} = (2\hat{i} + 3\hat{j}) \times 10^6$ m/s in a magnetic field $\vec{B} = (0.2\hat{i} - 0.3\hat{j} + 0.1\hat{k})$ T. Calculate the magnetic force in vector notation.
3.
A charged particle of mass $m$ and charge $q$ is accelerated through a potential difference $V$ and then enters a transverse magnetic field $B$. Prove that the radius of its circular path is given by $r = \sqrt{\frac{2mV}{qB^2}}$.
4.
Calculate the pitch of the helical path of an electron moving with a speed of $3 \times 10^7$ m/s at an angle of 30° to a uniform magnetic field of $1.5 \times 10^{-3}$ T. ($m_e = 9.1 \times 10^{-31}$ kg, $e = 1.6 \times 10^{-19}$ C).
5.
In a specific region, a uniform electric field $E = 3 \times 10^4$ V/m and magnetic field $B = 0.015$ T are mutually perpendicular. Find the velocity of an electron that passes through this region undeflected.
6.
A proton describes a circular path of radius 5 cm in a transverse magnetic field of 0.8 T. Calculate the kinetic energy of the proton in MeV. ($m_p = 1.67 \times 10^{-27}$ kg).
7.
Describe the exact trajectory of a charged particle if it enters a region containing both parallel electric and magnetic fields with an initial velocity perpendicular to the fields.
8.
An electron beam passes undeflected through mutually perpendicular $\vec{E}$ and $\vec{B}$ fields. If the electric field $\vec{E}$ is suddenly switched off, what will be the trajectory of the beam? Justify.
9.
A deuteron and an alpha particle are accelerated by the same accelerating potential difference and then enter a uniform magnetic field normal to the field lines. Find the ratio of their radii.
10.
Using the Lorentz force equation, explain why the kinetic energy of a charged particle moving in a purely uniform magnetic field remains strictly constant, even though its velocity vector changes continuously.
11.
A long straight wire carries a current of 50 A. An electron travels at $10^7$ m/s parallel to the wire at a distance of 0.1 m in the direction opposite to the current. Calculate the magnitude and direction of the magnetic force on the electron.
12.
Two concentric circular coils X and Y of radii 16 cm and 10 cm respectively, lie in the same vertical plane containing the N-S direction. Coil X has 20 turns and carries 16 A; coil Y has 25 turns and carries 18 A in the opposite sense. Find the magnitude of the net magnetic field at their common center.
13.
Use Biot-Savart's law to derive the mathematical expression for the magnetic field at a point on the central axis of a current-carrying circular loop.
14.
A straight wire carrying a current of 12 A is bent into a semi-circular arc of radius 2.0 cm. What is the magnetic field $\vec{B}$ exactly at the center of curvature of the arc?
15.
A wire placed precisely along the y-axis carries a steady current of 8 A in the +y direction. Calculate the magnetic field at point $P(0.04 \text{ m}, 0, 0)$.
16.
Derive the formula for the magnetic field at the exact center of a square current loop of side $a$ carrying a steady current $I$.
17.
A circular coil of 200 turns and radius 10 cm carries a current of 2 A. Calculate the magnetic field at a distance of 10 cm from its center along its axis.
18.
Two identical, long, straight wires intersect perpendicularly at the origin. Wire 1 lies along the x-axis carrying current $I$, and wire 2 lies along the y-axis carrying current $2I$. Find the magnetic field at point $(a, a, 0)$.
19.
Two identical uniform wires are shaped into a circle and a square respectively. If they carry the exact same current, find the ratio of the magnetic fields developed at their geometric centers.
20.
A straight conductor carrying current $I$ splits into two semi-circular arcs of unequal radii but re-joins to form a complete circular loop before continuing straight. Prove that the magnetic field at the center is zero if the incoming and outgoing straight wires are collinear.
21.
A long straight solid cylindrical wire of radius $a$ carries a steady current $I$ uniformly distributed across its cross-section. Derive an expression for the magnetic field at a distance $r < a$ and $r > a$ using Ampere's circuital law.
22.
An ideal solenoid of 60 cm length has 3 concentric layers of windings of 300 turns each. If the core radius is 2.0 cm and the current passed is 2.0 A, calculate the magnetic field near its center.
23.
A toroid has a core of inner radius 25 cm and outer radius 26 cm, around which 3500 turns of a wire are wound. If the current is 11 A, determine the magnetic field (i) exactly outside the toroid, (ii) deeply inside the core.
24.
A long coaxial cable consists of a solid inner cylindrical conductor of radius $R_1$ and an outer cylindrical shell of inner radius $R_2$ and outer radius $R_3$. The inner conductor carries a uniform current $I$ and the outer shell carries the same current $I$ in the opposite direction. Determine the magnetic field in the annular region $R_1 < r < R_2$.
25.
Does Ampere’s circuital law in its original form hold strictly true for a non-steady current (like a charging capacitor circuit)? Briefly explain why or why not.
26.
A solenoid of length 1.0 m and radius 2 cm has 1000 turns. Find the magnetic field exactly at one of its open ends if it carries a steady current of 5 A.
27.
Using the conceptual framework of Ampere's Law, mathematically prove why the magnetic field outside an ideal, infinitely long, tightly wound solenoid is considered to be zero.
28.
A thick copper wire of radius 5 mm carries a current of 10 A uniformly distributed over its cross-section. Find the magnitude of the magnetic field at a distance of exactly 2 mm from its central axis.
29.
Draw a detailed plot showing the variation of the magnitude of the magnetic field $B$ with distance $r$ from the central axis of a solid current-carrying cylinder of radius $R$, extending from $r=0$ to $r=3R$.
30.
If the number of turns in a toroid is doubled but the total current is halved, what will be the percentage change in the magnetic field inside the core of the toroid?
31.
Two long parallel straight wires A and B carry currents of 8.0 A and 5.0 A respectively in the same direction, separated by a distance of 4.0 cm. Estimate the net force acting on a 10 cm section of wire A.
32.
A rectangular loop of sides 25 cm and 10 cm carrying a current of 15 A is placed with its longer side parallel to a long straight wire carrying a current of 25 A. The straight wire is 2.0 cm away from the nearer long side of the loop. Calculate the net force on the rectangular loop.
33.
Three parallel wires P, Q, and R are placed coplanar with a separation of 2 cm between P and Q, and 2 cm between Q and R. They carry currents of 2 A, 4 A, and 6 A respectively in the same direction. Find the magnitude and direction of the net force per unit length on the middle wire Q.
34.
A square loop of side '$a$' carrying a current $I_2$ is kept at a distance '$x$' from an infinitely long straight wire carrying current $I_1$. Derive the mathematical expression for the net force on the square loop.
35.
Two infinitely long wires are mutually perpendicular and do not intersect (they are separated by a minimum distance $z$). Will there be any net translational magnetic force between them? Justify your answer.
36.
A 1.5 m long straight horizontal wire carries a current of 2 A and is suspended perfectly in mid-air by a uniform horizontal magnetic field of 0.5 T perpendicular to the wire. Find the mass of the wire. ($g = 10 \text{ m/s}^2$)
37.
Provide a complete mathematical and conceptual explanation showing why the magnetic force between two parallel wires carrying anti-parallel currents is inherently repulsive.
38.
A horizontal wire of length 0.1 m carries a current of 5 A. What minimum magnitude and specific direction of a uniform magnetic field can support the weight of this wire if its mass is $3 \times 10^{-3}$ kg?
39.
Two thin, exceptionally long parallel wires separated by distance '$d$' carry currents $I$ and $2I$ in the same direction. Determine the exact position where a third parallel wire should be placed so that it experiences absolutely zero net magnetic force.
40.
Define 1 Ampere legally using the concept of the magnetic force between two parallel current-carrying conductors. Why is this definition experimentally preferred over defining it via moving electrostatic charge?
41.
A circular coil of 20 turns and radius 10 cm is placed in a uniform magnetic field of 0.10 T normal to the plane of the coil. If the current in the coil is 5.0 A, compute the total torque and the total translational force on the coil.
42.
Derive the fundamental expression for the torque $\vec{\tau}$ acting on a rectangular current-carrying loop of area $A$, carrying current $I$, placed in a uniform magnetic field $B$.
43.
A square coil of side 10 cm consists of 20 turns and carries a current of 12 A. The coil is suspended vertically, and the normal to the plane of the coil makes an angle of 30° with the direction of a uniform horizontal magnetic field of 0.80 T. Calculate the magnitude of the torque experienced by the coil.
44.
Show mathematically that a planar current-carrying loop behaves analogously to a magnetic dipole. State the definitive expression for its magnetic dipole moment.
45.
Calculate the magnetic dipole moment of a revolving electron in the ground state of a hydrogen atom (radius $r = 0.53 \text{ Å}$, orbital velocity $v = 2.2 \times 10^6 \text{ m/s}$).
46.
A coil of area $0.05 \text{ m}^2$ having 500 turns is initially placed with its plane perpendicular to a magnetic field of $4 \times 10^{-5} \text{ T}$. It is rapidly rotated by 180° in 0.1 s. Calculate the total work done if the steady current in the coil is 1 A.
47.
Compare the magnetic dipole moment of a circular loop and a square loop if both are meticulously constructed from identical uniform wires of length $L$ and carry the exact same current $I$. Mathematically prove which geometry yields a greater moment.
48.
A magnetic dipole is placed in a uniform magnetic field. Define the precise vector conditions (angles) for it to exist in stable equilibrium and unstable equilibrium, respectively.
49.
Determine the potential energy of a current loop of magnetic moment $2.5 \text{ A}\cdot\text{m}^2$ when its area vector makes an angle of 120° with a uniform background magnetic field of 0.2 T.
50.
Elaborate conceptually on why a current loop placed in a *non-uniform* magnetic field experiences both a net translational force and a rotational torque simultaneously.
51.
Explain the core principle and the detailed working mechanism of a moving coil galvanometer, explicitly highlighting the role of the radial magnetic field.
52.
A galvanometer with a coil of resistance $12 \, \Omega$ shows a full-scale deflection for a current of 2.5 mA. How will you precisely convert it into an ammeter of range 0 to 7.5 A? Determine the net equivalent resistance of the resulting ammeter.
53.
A galvanometer has an internal resistance of $50 \, \Omega$ and requires 5 mA to produce a full-scale deflection. Calculate the exact series resistance required to convert it into a voltmeter reading up to 20 V.
54.
Define the 'figure of merit' of a galvanometer. State its formula and explain how it is inversely related to the current sensitivity of the instrument.
55.
Two moving coil galvanometers $M_1$ and $M_2$ have the following particulars: $R_1 = 10 \, \Omega$, $N_1 = 30$, $A_1 = 3.6 \times 10^{-3} \text{ m}^2$, $B_1 = 0.25 \text{ T}$ and $R_2 = 14 \, \Omega$, $N_2 = 42$, $A_2 = 1.8 \times 10^{-3} \text{ m}^2$, $B_2 = 0.50 \text{ T}$. Assuming identical restoring springs, determine the exact ratio of their current sensitivities and voltage sensitivities.
56.
Provide two distinct technical reasons why a soft iron cylinder is inserted as the core inside the coil of a moving coil galvanometer.
57.
An existing ammeter of resistance $0.80 \, \Omega$ can successfully measure current up to 1.0 A. What must be the precise value of an additional shunt resistance to enable this ammeter to measure current up to 5.0 A?
58.
A voltmeter has an internal resistance $G$ and an initial measuring range $V$. Mathematically calculate the resistance to be used in series with it to permanently extend its range to $nV$.
59.
Why must a high resistance be connected in series with a galvanometer to convert it into a voltmeter? Conceptually explain the negative circuit impact if a low resistance was used instead.
60.
Can a standard moving coil galvanometer be used directly to measure an alternating current (AC) circuit? Provide a rigorous justification for your answer.
