A rod of length 1 m rotates at 60 rad/s about one end in a field of 0.2 T parallel to the axis of rotation. The emf between its ends is
Answer: B
The emf is \( \tfrac{1}{2}B\omega l^2 \), that is \( 0.5 \times 0.2 \times 60 \times 1 = 6 \) V. The factor of one half arises because the velocity of a point on the rod grows linearly from zero at the axis, so its average is half the tip value.
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A transformer's secondary of 500 turns links a sinusoidal flux of 2 mWb peak at 50 Hz. The rms emf is about
Answer: C
The standard transformer equation is \( 4.44 f N \Phi_m \), which gives \( 4.44 \times 50 \times 500 \times 2 \times 10^{-3} = 222 \) V. The coefficient is \( 2\pi/\sqrt{2} \), combining the differentiation with the conversion to rms.
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True or false: the electric field in a time-varying situation is conservative.
Answer: D
A changing magnetic field gives the electric field a curl, so the line integral round a closed path is not zero and the field does net work round a loop. That is exactly the induced emf, and it is why a scalar potential alone will not do.
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True or false: the displacement current produces a magnetic field just as a conduction current does.
Answer: A
It enters Ampere's law on exactly the same footing as the conduction current, so it produces a magnetic field of the same kind and strength for the same number of amperes. The magnetic field between capacitor plates carrying alternating current is a real and measurable thing.
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True or false: Maxwell's equations require a medium to propagate a wave.
Answer: B
The free space constants describe the vacuum itself and the equations predict propagation with no material present at all. This was one of the most striking consequences of the theory and the experimental undoing of the aether.
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True or false: the tangential electric field is continuous across any interface.
Answer: C
It follows from Faraday's law applied to a vanishing rectangle, and no surface charge or current can affect it because the enclosed flux goes to zero with the area. It is the normal flux density and the tangential magnetic intensity that can jump.
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Assertion: Maxwell added the displacement current term to Ampere's law. Reason: without it the equations contradict the conservation of charge.
Answer: D
Taking the divergence of the unmodified law would force the current density to be solenoidal always, which fails whenever charge accumulates. The extra term makes the total current solenoidal instead, which is consistent, and that inconsistency is precisely what prompted the addition.
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Assertion: an emf appears in a stationary loop threaded by a changing flux. Reason: the charges in the loop experience a magnetic force because of their motion.
Answer: A
Nothing is moving, so the magnetic part of the Lorentz force plays no part; the driving agent is the induced electric field that a changing magnetic field creates. The reason describes the motional emf, which is the other and quite distinct mechanism.
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Assertion: the speed of light follows from the two free space constants. Reason: their product's reciprocal square root has the dimensions of a velocity.
Answer: B
The wave equation derived from Maxwell's equations has exactly that combination as its wave speed, and the dimensional check confirms it can be one. Substituting the measured values gives the measured speed of light, which was Maxwell's great result.
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Match the equation with what it states.
1. \( \nabla \cdot \mathbf{D} = \rho_v \) 2. \( \nabla \times \mathbf{E} = -\partial\mathbf{B}/\partial t \) 3. \( \nabla \cdot \mathbf{B} = 0 \)
P. No magnetic monopole exists Q. Charge is the source of electric flux R. A changing magnetic field induces an electric field
Answer: C
The four equations divide neatly into two statements about sources and two about how the fields generate one another. It is the second pair that makes waves possible, and the third equation's zero is what distinguishes magnetism from electricity.
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