Time Varying Fields and Maxwells Equations Questions and Answers

Q: A tiny oscillating magnetic dipole is formed by circulating sinusoidal current in a circular loop in the xz-plane, with center at origin. Consider this as a perfect dipole. Power radiated by the dipole is minimum along :

The radiated power is minimum along the y-axis.A current loop in the xz-plane has its magnetic dipole moment along the normal to the loop, the \( y \)-axis. Like an electric dipole, an oscillating magnetic dipole radiates with an intensity proportional to \( \sin^{2}\theta \), where \( \theta \) is measured from the dipole axis. So no power goes out along the axis itself.The \( x \)-axis, the \( z \)-axis and the direction \( x\hat{x} + z\hat{z} \) all lie in the plane of the loop, at \( \theta = 90^{\circ} \) from the moment, where the radiation is maximum.

Q: If J is the current density and ρ is the electric charge density, then equation \(\vec{\nabla}\cdot\vec{j} + \frac{\partial\rho}{\partial t} = 0\) represents :

The equation represents the continuity equation.It expresses conservation of charge. Integrated over a volume and using the divergence theorem, it says that the current flowing out through the surface equals the rate at which the charge inside decreases:\[ \oint \vec{j}\cdot d\vec{a} = -\frac{d}{dt}\int \rho\,dV \]The wave equation is second order in time and space, \(\nabla^{2}\psi = \frac{1}{v^{2}}\frac{\partial^{2}\psi}{\partial t^{2}}\). Poisson's equation, \(\nabla^{2}\varphi = -\rho/\varepsilon_{0}\), relates potential to charge density, and Laplace's equation is its charge-free form. Neither involves current density.

Q: What happens to the current in a coil while accelerating a magnet inside it ?

The current in the coil increases.By Faraday's law the induced emf is proportional to the rate of change of flux linkage, \(e = -N\,d\phi/dt\). If the magnet moves through the coil at a steady speed, the flux changes at a steady rate and the emf and current are roughly constant. If the magnet is accelerating, its speed keeps rising, so the flux changes faster and faster and the induced emf, and hence the current in a closed coil, keeps increasing.The current would decrease if the magnet were slowing down, would be constant at steady speed, and reverses only when the magnet's motion reverses or it passes out of the far end; acceleration in one direction does none of these.

Q: Which of the following laws does not form a Maxwell equation ?

The law that does not form a Maxwell equation is Planck's law.Maxwell's four equations are Gauss's law for electricity, Gauss's law for magnetism (no magnetic monopoles), Faraday's law of induction, and Ampere's circuital law with Maxwell's displacement current added. Together they describe all classical electromagnetic fields and predict electromagnetic waves.Planck's law describes the spectrum of black-body radiation and introduced the quantisation of energy, \(E = h\nu\). It belongs to thermal radiation and quantum physics and is not one of Maxwell's equations.

81.

A tiny oscillating magnetic dipole is formed by circulating sinusoidal current in a circular loop in the xz-plane, with center at origin. Consider this as a perfect dipole. Power radiated by the dipole is minimum along :

A.x-axis
B.y-axis
C.z-axis
D.\( x\hat{x} + z\hat{z} \)

Exam Year(s)

MH SET - 2018, SET Exam - 2018,

Answer: B

The radiated power is minimum along the y-axis.

A current loop in the xz-plane has its magnetic dipole moment along the normal to the loop, the \( y \)-axis. Like an electric dipole, an oscillating magnetic dipole radiates with an intensity proportional to \( \sin^{2}\theta \), where \( \theta \) is measured from the dipole axis. So no power goes out along the axis itself.

The \( x \)-axis, the \( z \)-axis and the direction \( x\hat{x} + z\hat{z} \) all lie in the plane of the loop, at \( \theta = 90^{\circ} \) from the moment, where the radiation is maximum.

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82.

If J is the current density and ρ is the electric charge density, then equation \(\vec{\nabla}\cdot\vec{j} + \frac{\partial\rho}{\partial t} = 0\) represents :

A.wave equation
B.continuity equation
C.Poisson's equation
D.Laplace's equation

Exam Year(s)

MH SET - 2023, SET Exam - 2023,

Answer: B

The equation represents the continuity equation.

It expresses conservation of charge. Integrated over a volume and using the divergence theorem, it says that the current flowing out through the surface equals the rate at which the charge inside decreases:

\[ \oint \vec{j}\cdot d\vec{a} = -\frac{d}{dt}\int \rho\,dV \]

The wave equation is second order in time and space, \(\nabla^{2}\psi = \frac{1}{v^{2}}\frac{\partial^{2}\psi}{\partial t^{2}}\). Poisson's equation, \(\nabla^{2}\varphi = -\rho/\varepsilon_{0}\), relates potential to charge density, and Laplace's equation is its charge-free form. Neither involves current density.

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83.

What happens to the current in a coil while accelerating a magnet inside it ?

A.Increases
B.Decreases
C.Remains constant
D.Reverses

Exam Year(s)

MH SET - 2024, SET Exam - 2024,

Answer: A

The current in the coil increases.

By Faraday's law the induced emf is proportional to the rate of change of flux linkage, \(e = -N\,d\phi/dt\). If the magnet moves through the coil at a steady speed, the flux changes at a steady rate and the emf and current are roughly constant. If the magnet is accelerating, its speed keeps rising, so the flux changes faster and faster and the induced emf, and hence the current in a closed coil, keeps increasing.

The current would decrease if the magnet were slowing down, would be constant at steady speed, and reverses only when the magnet's motion reverses or it passes out of the far end; acceleration in one direction does none of these.

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84.

The ratio of conduction to displacement current density is referred to as :

A.Attenuation constant
B.Propagation constant
C.Loss tangent
D.Dielectric constant

Exam Year(s)

MH SET - 2024, SET Exam - 2024,

Answer: C

The ratio of conduction to displacement current density is the loss tangent.

In a medium with conductivity \(\sigma\) and permittivity \(\varepsilon\), a time-harmonic field drives a conduction current density \(\sigma E\) and a displacement current density \(j\omega\varepsilon E\). Their magnitudes give

\[ \begin{aligned} \tan\theta &= \frac{|J_{c}|}{|J_{d}|} \\ &= \frac{\sigma}{\omega\varepsilon} \end{aligned} \]

which is the loss tangent. It classifies media: much greater than 1 for a good conductor, much less than 1 for a good dielectric.

The attenuation constant \(\alpha\) and propagation constant \(\gamma\) describe how a wave decays and travels, and the dielectric constant is \(\varepsilon_{r}\) alone; they depend on the loss tangent but are not that ratio.

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85.

Which of the following laws does not form a Maxwell equation ?

A.Planck's law
B.Gauss's Law
C.Faraday's law
D.Ampere's Law

Exam Year(s)

MH SET - 2025, SET Exam - 2025,

Answer: A

The law that does not form a Maxwell equation is Planck's law.

Maxwell's four equations are Gauss's law for electricity, Gauss's law for magnetism (no magnetic monopoles), Faraday's law of induction, and Ampere's circuital law with Maxwell's displacement current added. Together they describe all classical electromagnetic fields and predict electromagnetic waves.

Planck's law describes the spectrum of black-body radiation and introduced the quantisation of energy, \(E = h\nu\). It belongs to thermal radiation and quantum physics and is not one of Maxwell's equations.

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