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PhysicsElectromagnetic Waves
The electric field in a plane electromagnetic wave is given by Ez = 60 cos(5x + 1.5 x 10^9 t) V/m. Then the expression for the corresponding magnetic field is:
Options
1
By = 60 sin(5x + 1.5 x 10^9 t) T
2
By = 2 x 10^-7 cos(5x + 1.5 x 10^9 t) T
3
Bx = 2 x 10^-7 cos(5x + 1.5 x 10^9 t) T
4
Bz = 60 cos(5x + 1.5 x 10^9 t) T
Correct Answer
By = 2 x 10^-7 cos(5x + 1.5 x 10^9 t) T
Solution
1

Wave: Ez in z-direction, propagates in x-direction (kx+wt means -x direction).

B must be perpendicular to both E(z) and propagation(x) so B is in y-direction. Bx and Bz are eliminated.

2

Magnitude: B0 = E0/c = 60/(3x10^8) = 2x10^-7 T

Phase: E and B always in same phase, so cosine (not sine). Option 1 (sin) is wrong.

Answer: By = 2x10^-7 cos(5x + 1.5x10^9 t) T

B0 = E0/c = 2x10^-7 T | Direction: y (perp to E in z and propagation in x) | Phase: cosine = same as E
Theory: Electromagnetic Waves
1. EM Wave Properties

In a plane EM wave: E and B are mutually perpendicular and both perpendicular to propagation direction. They are in the same phase (both reach peak and zero simultaneously). Ratio E0/B0 = c = 3x10^8 m/s always. Direction of propagation = E x B. Speed = omega/k = c in vacuum. The wave can travel through vacuum without any medium.

2. Maxwell Equations

Maxwell added displacement current Id = epsilon0 (d phi_E/dt) to Ampere law. This predicted EM waves. Faraday law: changing B creates E. Ampere-Maxwell law: changing E creates B. Together they sustain EM wave propagation. Verified experimentally by Hertz in 1887.

3. EM Spectrum

From lowest to highest frequency: Radio waves, Microwaves, Infrared, Visible (VIBGYOR 400-700 nm), Ultraviolet, X-rays, Gamma rays. All travel at c = 3x10^8 m/s in vacuum. Higher frequency = higher photon energy E = hf.

4. Intensity and Energy

Intensity = Poynting vector magnitude S = E0B0/(2mu0) = epsilon0 c E0^2 / 2. Energy equally distributed between E and B. Radiation pressure = I/c (absorption) or 2I/c (reflection). Solar constant = 1361 W/m^2.

5. Polarization

EM waves are transverse so can be polarized. Malus law: I = I0 cos^2(theta). Brewster angle: tan(thetaB) = n. At Brewster angle reflected light is completely polarized. Polaroids produce linear polarization by selective absorption of one component.

6. Displacement Current

When capacitor is being charged, no real current flows in gap but displacement current Id = epsilon0 dE/dt flows. This makes total current continuous. Physically: changing E creates B. This enables EM wave propagation in vacuum.

7. Wave Speed Calculation

For wave E = E0 cos(kx + wt): speed = w/k. Here w = 1.5x10^9 rad/s, k = 5 rad/m. Speed = 1.5x10^9/5 = 3x10^8 m/s = c. Wavelength = 2pi/k = 1.26 m. Frequency = w/2pi = 2.39x10^8 Hz. The +wt term means wave travels in -x direction.

8. Applications

Radio/TV: AM 530-1600 kHz, FM 88-108 MHz. Microwave ovens: 2.45 GHz resonates water molecules. RADAR: pulse timing gives distance. Medical X-rays: bone imaging. Gamma rays: cancer radiotherapy. Infrared: remote controls, thermal cameras, optical fibre. UV: vitamin D synthesis, sterilization.

Frequently Asked Questions
1. Why By and not Bx or Bz? ⌄
E is in z, propagation in x. B must be perpendicular to both: z x B must give x-direction (propagation). z x (-y) = x. So B in -y direction, or for this wave orientation, B in y-direction. Neither x nor z satisfies the perpendicularity requirement.
2. Why E0/B0 = c? ⌄
This comes from Maxwell equations. In EM wave: dE/dx = -dB/dt and dB/dx = -mu0 epsilon0 dE/dt. Solving these gives speed = 1/sqrt(mu0 epsilon0) = c and the ratio E0/B0 = c as a necessary consequence.
3. What is displacement current and why is it important? ⌄
Displacement current Id = epsilon0 dE/dt fills the gap in capacitor where no real current flows during charging. Maxwell added this to Ampere law to make it consistent for non-static situations. Without it, Ampere law gives contradictory results for different surfaces surrounding same circuit. With it, total current is always continuous, and it predicts existence of EM waves.
4. How does an EM wave carry energy and momentum? ⌄
EM waves carry energy via the Poynting vector S = (1/mu0)(E x B) in W/m^2. They also carry momentum p = U/c per unit volume (U = energy density). This radiation pressure can push objects (solar sails). Light exerts tiny but measurable pressure. Laser tweezers use this to trap and manipulate tiny particles.
5. What is the difference between plane wave and spherical wave? ⌄
Plane wave: wavefronts are infinite flat planes, intensity stays constant, E and B uniform across each wavefront. Example: laser beam approximated as plane wave far from source. Spherical wave: wavefronts are spheres expanding from a point source, intensity decreases as 1/r^2 (inverse square law). Example: light from a star or lamp. Real EM waves from antennas or stars are spherical but appear planar far from source.
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