CUET UG Physics Booster Test 2-Dipole Dynamics and Electrostatic Analogs

When a magnetic dipole of magnetic moment m is placed in a uniform magnetic field B, it experiences a torque given by: τ=m×B The magnitude of the torque is: τ=mBsin⁡θ where θ is the angle between the magnetic moment and the magnetic field. This torque tends to rotate the dipole so that its magnetic moment aligns with the external magnetic field. Therefore, it acts as a restoring torque whenever the dipole is displaced from its stable equilibrium position. The torque becomes maximum at θ = 90° and becomes zero when the dipole is parallel or anti-parallel to the field. Since the torque arises from a vector cross product, it is itself a vector quantity and not a scalar. Therefore, Option C correctly describes the role of magnetic torque.

The torque experienced by a magnetic dipole in a uniform magnetic field is: τ=mBsin⁡θ The maximum torque occurs when: θ=90^∘ because: sin⁡90^∘=1 Therefore: τ_(max)=mB Substituting the given values: τ_(max)=0.32×0.15τ_(max)=0.048 N m Thus, the maximum restoring torque acting on the magnet is: 0.048 N m Unit Verification τ=(J T^(-1))(T)=J=N m Hence the unit is correct.

The torque acting on a magnetic dipole in a uniform magnetic field is: τ=mBsin⁡θ Potential energy is obtained from the work done against this restoring torque: U=-∫τdθ Substituting the torque expression: U=-∫mBsin⁡θ dθU=-mBcos⁡θ+C NCERT chooses the reference level of potential energy at: θ=90^∘ At this angle: U=0 which makes the constant of integration zero. Thus: U=-mBcos⁡θ or U=-m⋅B Therefore Statements 1, 2 and 4 are correct.

The magnetic potential energy of a dipole in a uniform magnetic field is obtained by integrating the restoring torque. The resulting expression is: U_m=-mBcos⁡θ Since: m⋅B=mBcos⁡θ the expression becomes: U_m=-m⋅B This equation shows that magnetic potential energy depends on both the magnitude of the magnetic moment and the angle between the dipole and the magnetic field. The energy is minimum when the dipole is parallel to the field and maximum when it is anti-parallel. Because potential energy is a scalar quantity, it must be represented using a dot product rather than a cross product.

The potential energy of a magnetic dipole in a magnetic field is: U=-mBcos⁡θ For θ = 0°: U=-mB which is the minimum value and corresponds to stable equilibrium. For θ = 90°: U=0 For θ = 180°: U=+mB which is the maximum value and corresponds to unstable equilibrium. For θ = 45°: U=-mBcos⁡45^∘U=-mB/√(2) Therefore: 1 → c 2 → d 3 → b 4 → a Hence the correct matching is Option A.

The potential energy of a magnetic dipole in a uniform magnetic field is given by: U=-mBcos⁡θ For unstable equilibrium: θ=180^∘ Since: cos⁡180^∘=-1 the potential energy becomes: U=+mB which is the maximum possible value. Therefore, the dipole is in unstable equilibrium. In this position, the magnetic moment is anti-parallel to the magnetic field. A slight angular displacement from this position produces a torque that moves the dipole away from the unstable position and ultimately towards the stable equilibrium position where θ = 0°. Thus, Statements 1, 3 and 4 are correct, while Statement 2 is incorrect because unstable equilibrium corresponds to maximum, not minimum, potential energy.

NCERT shows a close analogy between electric dipoles and magnetic dipoles. Many magnetic field expressions can be obtained directly from the corresponding electric field expressions by replacing certain quantities. The electric dipole moment: p is replaced by the magnetic dipole moment: m Similarly, the electrostatic constant: 1/ε_0 is replaced by: μ_0 where μ₀ is the permeability of free space. These substitutions allow the mathematical structure of electric dipole equations to be converted into magnetic dipole equations. Therefore, the correct replacements are m and μ₀.

The mathematical analogy between electric dipoles and magnetic dipoles is one of the most important concepts discussed in NCERT magnetism. The correspondence is: E↔Bp↔m1/ε_0↔μ_01/4πε_0↔μ_0/4π These substitutions allow electric dipole field equations to be transformed into magnetic dipole field equations. Statement D is incorrect because μ₀ is explicitly introduced as the magnetic counterpart of the electrostatic factor 1/ε₀. Therefore, μ₀ is not independent of the analogy; it is an essential part of it.

For a short magnetic dipole: Axial field: B_A=μ_0/4π2m/r^3 Equatorial field: B_E=μ_0/4πm/r^3 Taking the ratio: B_A/B_E=(μ_0/4π2m/r^3)/(μ_0/4πm/r^3)B_A/B_E=2 Thus, the axial magnetic field is twice the magnitude of the equatorial magnetic field at the same distance from the dipole.

For a short magnetic dipole, the magnetic field on the axial line is given by: B_A=μ_0/4π2m/r^3 The field vector is directly proportional to the magnetic moment vector. Since there is no negative sign in the axial field expression, the field direction remains the same as the direction of the magnetic moment vector. Therefore, along the axial line, the magnetic field points from the south pole towards the north pole outside the magnet. This is in contrast to the equatorial field, where the field direction is opposite to the magnetic moment vector because of the negative sign in the corresponding expression.

For a short bar magnet treated as a magnetic dipole, the magnetic field at a point on the equatorial line is given by BE = -(μ₀/4π)(m/r³) where m is the magnetic dipole moment and r is the distance from the centre of the magnet. The negative sign is extremely important because it indicates that the equatorial magnetic field is directed opposite to the magnetic moment vector. Physically, the magnetic moment vector points from the south pole to the north pole inside the magnet. At points on the equatorial line, the resultant magnetic field is directed opposite to this vector. This result is obtained by resolving the contributions of the two poles and applying the dipole approximation (r >> l). NCERT highlights that the axial field and equatorial field differ both in magnitude and direction.

For a magnetic dipole, the magnetic field on the axial line is BA = (μ₀/4π)(2m/r³) and is directed along the magnetic moment vector m. In contrast, the magnetic field on the equatorial line is BE = -(μ₀/4π)(m/r³) which is directed opposite to m. The negative sign explicitly indicates this anti-parallel relationship. Externally, magnetic field lines emerge from the north pole and enter the south pole. At points on the equatorial line, the resultant field therefore points from north to south, opposite to the magnetic moment direction. These directional relationships are fundamental in understanding dipole fields and are repeatedly used while analyzing magnetic interactions and potential energy of dipoles.

The torque on an electric dipole placed in an external electric field is given by τ = p × E where p is the electric dipole moment and E is the electric field. Similarly, the torque on a magnetic dipole placed in an external magnetic field is τ = m × B where m is the magnetic dipole moment and B is the magnetic field. These expressions are direct analogues of one another and demonstrate the close similarity between electrostatics and magnetism. The torque acts to rotate the dipole so that its dipole moment aligns with the external field. When the dipole becomes parallel to the field, the torque becomes zero and the system reaches equilibrium.

The potential energy of an electric dipole in an external electric field is U = -p·E where p is the electric dipole moment and E is the electric field. Similarly, the potential energy of a magnetic dipole in an external magnetic field is U = -m·B Both expressions involve scalar (dot) products and therefore produce scalar quantities. The negative sign indicates that the energy is minimum when the dipole moment aligns with the external field. These expressions are fundamental results in the analogy between electric and magnetic dipoles and are extensively used in equilibrium and stability problems.

The interaction energy between two magnetic dipoles arises because one dipole is placed in the magnetic field produced by the other. This energy is given by U = -m·B which clearly depends on the relative orientation between the magnetic moment and the magnetic field. Therefore, changing the orientation of either dipole changes the potential energy of the system. If dipole Q lies on the normal bisector of dipole P, the magnetic field at Q is calculated using the equatorial field expression. If Q lies on the axis of P, the axial field formula must be used. Since potential energy depends strongly on orientation, the statement claiming independence from relative orientation is incorrect.

The potential energy of a magnetic dipole placed in an external magnetic field is given by U = -m·B = -mB cos θ where m is the magnetic dipole moment, B is the magnetic field and θ is the angle between them. A system is in stable equilibrium when its potential energy is minimum. On the perpendicular bisector (equatorial line) of dipole P, the magnetic field is directed opposite to the magnetic moment of P. For dipole Q to remain in stable equilibrium, it must orient itself parallel to the local magnetic field so that θ = 0°. In this case, U = -mB which is the minimum possible value. Since the potential energy is minimum and negative, the equilibrium is stable. Any small displacement increases the potential energy and produces a restoring torque that returns the dipole to equilibrium.

A magnetic dipole placed in a uniform magnetic field experiences a torque that tends to align the dipole with the field, but the forces on the two poles are equal and opposite, resulting in zero net force. In a non-uniform magnetic field, the magnetic forces on the poles are unequal, producing both torque and a net force. An iron nail near a bar magnet becomes magnetised by induction, developing an induced magnetic moment that experiences both attraction and torque. When a bar magnet is cut into two pieces, each piece still contains both a north pole and a south pole; isolated magnetic monopoles are not produced. These principles form the basis for understanding magnetic interactions in practical systems.

When an iron nail is brought near a bar magnet, the magnetic field of the magnet induces magnetisation in the nail. The end of the nail closer to the north pole of the magnet develops an induced south pole, while the farther end develops an induced north pole. Because the magnetic field near the magnet is non-uniform, the attractive force between the nearby opposite poles is stronger than the repulsive force between the distant like poles. As a result, the nail experiences a net attractive force toward the magnet. In addition, the induced magnetic moment interacts with the magnetic field and can experience torque that tends to align the nail along the field direction. Therefore, Statements 1, 2 and 4 are correct.

A magnetic field line is an imaginary curve whose tangent at any point gives the direction of the magnetic field at that location. A small compass needle placed at the same point aligns itself along the local magnetic field direction. Consequently, the axis of the compass needle becomes parallel to the tangent drawn to the magnetic field line. This property is used experimentally to map magnetic field patterns around magnets and current-carrying conductors. The direction indicated by the north pole of the compass corresponds to the direction of the magnetic field vector. Therefore, both the tangent to the magnetic field line and the compass needle point in the same direction and are parallel to each other.

Different elements of a current-carrying conductor produce magnetic fields that can act on other elements of the same conductor. Thus, internal magnetic forces can exist within a wire. For a perfectly straight wire, symmetry ensures that the resultant internal magnetic force on the wire is zero. However, in curved conductors such as circular loops, different current elements exert forces on one another, producing internal mechanical stresses. These stresses are responsible for the tendency of a current loop to expand slightly. A current element does not exert a magnetic force or torque on itself because self-interaction is not considered in the classical treatment of magnetic forces. Therefore, Statements 1, 2 and 4 are correct, while Statement 3 is incorrect.