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The Vault
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The Vault
CBSE Class 12 Physics · 11 questions · 26 marks
Maxwell's four equations quietly predicted something nobody had asked for: that a changing electric field alone could create a magnetic field, closing a loop that lets light travel through empty space without any medium at all. This short but conceptually dense chapter introduces displacement current, shows how it fixes a gap in Ampere's law, and tours the electromagnetic spectrum from radio waves to gamma rays.
The concept of displacement current was introduced by Maxwell to explain the magnetic field that exists:
Answer
In the region between the plates of a charging capacitor, where no conduction current flows — Ampere's original law failed for a capacitor's gap, since different surfaces bounded by the same loop gave inconsistent enclosed currents. Maxwell added the displacement current I_d = ε₀ dΦ_E/dt, due to the changing electric field there, restoring consistency and predicting a magnetic field in that gap.
In a plane electromagnetic wave travelling through vacuum, the electric field and magnetic field are:
Answer
Perpendicular to each other and both perpendicular to the direction of propagation — an electromagnetic wave is transverse; E and B oscillate in phase, mutually at right angles, and both lie in the plane perpendicular to the direction of travel, which is given by the direction of E × B.
Which of the following correctly orders these electromagnetic radiations from smallest to largest wavelength?
Answer
Gamma rays, visible light, infrared, radio waves — this list runs from the shortest wavelength (gamma rays) through visible light and infrared to the longest wavelength (radio waves), which is the correct increasing order. Wavelength and frequency are inversely related across the spectrum, since all these waves travel at the same speed c in vacuum.
The amplitude of the magnetic field in a plane electromagnetic wave is 2.0×10⁻⁸ T. The amplitude of the electric field is closest to:
Answer
6.0 V/m — in vacuum the field amplitudes are related by E₀ = cB₀. E₀ = 3×10⁸ × 2.0×10⁻⁸ = 6.0 V/m. This fixed ratio holds at every instant for a plane electromagnetic wave in free space.
Assertion (A): Electromagnetic waves do not require a material medium for their propagation. Reason (R): Electromagnetic waves consist of oscillating electric and magnetic fields that regenerate each other according to Maxwell's equations, without relying on the vibration of any material particles.
Answer
Both A and R are true and R is the correct explanation of A — a changing E field produces a B field (Ampere-Maxwell law) and a changing B field produces an E field (Faraday's law); this mutual regeneration lets the disturbance propagate through vacuum, unlike mechanical waves such as sound, which need particles of a medium to vibrate.
A parallel plate capacitor with circular plates of radius 5.0 cm is being charged, so that the electric field between the plates changes at a rate of 2.0×10¹² V m⁻¹ s⁻¹. Calculate the displacement current between the plates.
Answer
Displacement current, I_d = ε₀ (dΦ_E/dt) = ε₀ A (dE/dt), since Φ_E = EA for a uniform field. A = πr² = 3.14 × (0.050)² = 3.14 × 2.5×10⁻³ = 7.85×10⁻³ m². I_d = 8.85×10⁻¹² × 7.85×10⁻³ × 2.0×10¹² I_d = 8.85×10⁻¹² × 1.57×10¹⁰ I_d = 0.139 A ≈ 0.14 A. This displacement current equals the conduction current flowing in the connecting wires at that same instant.
State three properties common to all electromagnetic waves and explain why microwaves, rather than radio waves, are used in radar systems.
Answer
Three common properties: (i) all electromagnetic waves travel at the same speed c = 3×10⁸ m/s in vacuum, regardless of wavelength; (ii) they are transverse waves, with E and B mutually perpendicular and both perpendicular to the direction of propagation; (iii) they carry energy and momentum and can travel through vacuum without needing a medium. Microwaves are preferred for radar because their short wavelength (centimetre range) allows them to be focused into narrow, directional beams using antennas of a practical size, giving good angular resolution for locating objects; their frequency also matches convenient generation by devices such as klystrons and magnetrons, unlike the much longer, harder-to-focus wavelengths of ordinary radio waves.
Name the part of the electromagnetic spectrum used in each of the following, and give one reason for the choice: (i) sterilising surgical instruments, (ii) taking images of bone fractures.
Answer
(i) Ultraviolet radiation is used to sterilise surgical instruments, because its photon energy is high enough to damage the DNA and proteins of bacteria and other microorganisms, killing them without the material-penetrating hazards of X-rays. (ii) X-rays are used for imaging bone fractures, because their short wavelength gives them high penetrating power through soft tissue while being absorbed more strongly by the denser calcium in bone, producing sharp contrast on the image.
Explain how Maxwell modified Ampere's circuital law by introducing the concept of displacement current, illustrating the inconsistency it resolves using the example of a charging capacitor. State the complete Ampere-Maxwell law.
Answer
The inconsistency: Ampere's original law states ∮B·dl = μ₀I, where I is the current enclosed by any surface bounded by the loop over which the line integral is taken. For a circuit charging a capacitor, take an Amperian loop encircling the connecting wire. If a flat surface bounded by this loop is chosen, it is pierced by the conduction current I in the wire, giving ∮B·dl = μ₀I. But a second surface bounded by the same loop can instead be drawn bulging through the gap between the capacitor plates, where no conduction current flows at all (the plates are separated by an insulating gap). This surface gives ∮B·dl = 0, contradicting the first result for the very same loop. Maxwell's resolution: he recognised that although no charge crosses the gap, the electric field between the plates is changing as charge accumulates, and proposed that this changing electric flux acts as an additional source of magnetic field, exactly compensating for the missing conduction current. He defined the displacement current as I_d = ε₀ (dΦ_E/dt), where Φ_E is the electric flux through the surface in question. Between the plates, I_d equals the actual conduction current I flowing in the wires at every instant, so both choices of surface now give the same enclosed current. The complete Ampere-Maxwell law is ∮B·dl = μ₀(I_c + I_d) = μ₀I_c + μ₀ε₀ (dΦ_E/dt), which holds consistently for every surface bounded by the loop, and which, together with Faraday's law, allowed Maxwell to predict the existence of self-sustaining electromagnetic waves.
(a) Starting from the relation E₀ = cB₀ and the formula for energy density of electric and magnetic fields, show that the average energy density of an electromagnetic wave is shared equally between its electric and magnetic parts. (b) List the electromagnetic spectrum in order of increasing frequency, naming at least six bands, and give one practical use of microwaves and one of gamma rays.
Answer
(a) The instantaneous energy density stored in the electric field is u_E = ½ε₀E², and in the magnetic field u_B = B²/2μ₀. Using E = cB and c = 1/√(μ₀ε₀), so c² = 1/(μ₀ε₀): u_B = B²/2μ₀ = (E/c)²/2μ₀ = E²/(2μ₀c²) = E² × μ₀ε₀/(2μ₀) = ½ε₀E². So u_B = u_E at every instant, which means the two forms of energy density are exactly equal, not just on average. Averaged over a cycle, each contributes half the total energy density of the wave, u = u_E + u_B = ε₀E² (instantaneous), with time-average ⟨u⟩ = ε₀E_rms² split equally between the electric and magnetic parts. (b) In order of increasing frequency: radio waves, microwaves, infrared, visible light, ultraviolet, X-rays, gamma rays. Microwaves are used in radar systems and in microwave ovens, where their frequency matches the rotational resonance of water molecules, heating food efficiently. Gamma rays are used in the radiotherapy treatment of cancer, where their high photon energy destroys malignant cells, and in sterilising medical equipment.
Read the passage and answer the questions that follow: A weather satellite in orbit transmits data to a ground station using a microwave signal of frequency 4.0 GHz (4.0×10⁹ Hz). Engineers need to know how the signal will behave to design the receiving antenna and estimate transmission delay. (a) Calculate the wavelength of this microwave signal in vacuum. (b) The satellite is at a height of 36000 km above the ground station. Calculate the time taken for the signal to reach the ground, assuming it travels the straight-line distance in vacuum. (c) State whether this radiation is ionising or non-ionising, and give a reason based on its position in the spectrum. (d) Explain briefly why microwaves, rather than visible light, are chosen for this satellite link even though both are electromagnetic waves travelling at the same speed.
Answer
(a) λ = c/ν = 3×10⁸/4.0×10⁹ = 7.5×10⁻² m = 7.5 cm. (b) t = distance/speed = 36000×10³/3×10⁸ = 3.6×10⁷/3×10⁸ = 0.12 s, that is 120 milliseconds. (c) Microwaves are non-ionising radiation. Being at the low-frequency end of the spectrum, their photon energy (E = hν) is far too small to knock electrons out of atoms or break chemical bonds, unlike the high-frequency ultraviolet, X-ray and gamma-ray regions. (d) Microwaves are chosen because they pass through the atmosphere (including clouds) with little absorption or scattering and can be generated and detected efficiently by antennas of a convenient size and directed as narrow beams; visible light, in contrast, is heavily scattered by clouds and atmospheric particles, and is blocked by any obstruction, making it unreliable for a continuous satellite link, even though its speed in vacuum is identical.
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