Home › Physics › Electricity and Magnetism › Transverse Nature of Electromagnetic Waves
Electricity and Magnetism
Transverse Nature of Electromagnetic Waves
Electromagnetic waves propagate through space via mutually perpendicular, oscillating electric and magnetic field vectors that are also perpendicular to the direction of wave propagation. This intrinsic vector orientation confirms their purely transverse wave character.
UP TGT ScienceUP PGT PhysicsStudy Notes + Practice
Scope and prerequisites
TGT core
Understanding the direction of electric field E, magnetic field B, and wave velocity c, along with basic right-hand rule applications and electromagnetic spectrum properties.
PGT extension
Rigorous derivation from Maxwell's equations in free space, phase relationships between E and B, impedance of free space, and mathematical proof of transverse nature using divergence equations.
Prerequisites
Maxwell's equations in differential and integral forms
Vector calculus identities and cross products
Properties of plane harmonic waves
Orthogonality of E, B, and Propagation Vector k
The electric field vector (E) and magnetic field vector (B) in an electromagnetic wave oscillate perpendicular to each other.
Both fields are strictly perpendicular to the direction of propagation given by the wave vector (k) or pointing vector direction.
The cross product E x B points precisely in the direction of energy propagation, which is the direction of the wave velocity c.
Proof via Maxwell's Equations in Free Space
In charge-free and current-free space, Maxwell's curl equations govern the spatial and temporal variations of E and B.
Applying Faraday's law of induction (curl $E = -dB/dt) forces the electric field$ to be perpendicular to the wave propagation vector.
Applying Ampere-Maxwell law (curl $B = \mu_0 \epsilon_0 dE/dt) forces the magnetic field$ to be perpendicular to the wave propagation vector.
Magnitude Ratio and Impedance of Free Space
The magnitude of the electric field and magnetic field are related by the constant wave speed: E = cB.
The ratio of the magnitudes of electric to magnetic field amplitudes equals the characteristic impedance of free space ($Z_0 = \sqrt(\mu_0 / \epsilon_0))$.
The numerical value of the characteristic impedance of free space is approximately 376.7 ohms.
Polarization and Transverse Verification
The transverse nature allows electromagnetic waves to exhibit polarization, where the electric field vector is restricted to a specific plane.
Passage through polaroids demonstrates extinction when transmission axes are perpendicular, confirming the transverse vector model.
Longitudinal electromagnetic waves do not exist in free space because zero divergence conditions rule out parallel field oscillations.
Derivations
Derivation of Transverse Nature from Maxwell's Equations
Conditions: Free space (rho = 0, J = 0), monochromatic plane wave traveling in z-direction
Assume a plane wave solution where fields depend only on z and t: E(z,t) = E_0 exp(i(kz - omega t)) and B(z,t) = B_0 exp(i(kz - omega t)).
Evaluate Faraday's law: curl $E = - \partial B / \partial t$. The curl operation for a z-dependent field yields (partial E_y / partial z) hat{x} - (partial E_x / partial z) hat{y} = - partial B / partial t.
Equating components shows that partial E_z / partial z = 0, which implies the longitudinal component of E has no spatial wave dependence and can be set to zero for traveling waves.
Similarly, apply Ampere-Maxwell law: curl $B = \mu_0 \epsilon_0 \partial E / \partial t$, which establishes that $B_z = 0$.
Conclusion: Both E_z and B_z vanish along the direction of propagation (z-axis), proving that E and B have only transverse components (x and y).
Result: $E_z = 0$ and $B_z = 0$, proving the purely transverse nature of electromagnetic waves.
Derivation of Wave Equation and Field Amplitude Ratio
Conditions: Free space electromagnetic wave propagation
Take the curl of Faraday's equation: curl(curl E) = - partial / partial t (curl B).
Use the vector identity curl(curl E) = grad(div E) - nabla^2 E. Since div E = 0 in free space, this simplifies to - nabla^2 E.
Substitute Ampere-Maxwell equation for curl B: - nabla^2 $E = - \mu_0 \epsilon_0 \partial^2 E / \partial t^2$.
Obtain the standard wave equation: nabla^2 $E = \mu_0 \epsilon_0 \partial^2 E / \partial t^2$, identifying wave speed $c = 1 / \sqrt(\mu_0 \epsilon_0)$.
Substitute harmonic forms into Faraday's equation to find k $E_0 = \omega B_0$, leading to E_0 / $B_0 = \omega / k = c$.
S = energy flux density vector, E = electric field, B = magnetic field, m$u_0 = permeability$ W/m^2
Instantaneous energy flow
Harmonic Electric Field Wave
$$E(z,t) = E_0 \cos(kz - \omega t)$$
$k = propagation constant (2pi/\lambda)$, omega = angular frequency V/m
Wave traveling along positive z-axis
Harmonic Magnetic Field Wave
$$B(z,t) = B_0 \cos(kz - \omega t)$$
$B_0 = magnetic field amplitude$, k = wave number, omega = angular frequency Tesla
Wave traveling along positive z-axis, in phase with E
Relation between k and omega
$$v = \frac{\omega}{k} = c$$
omega = angular frequency, k = angular wave number m/s
Free space wave propagation
Intensity of Electromagnetic Wave
$$I = \frac{1}{2} c \epsilon_0 E_0^2$$
I = average intensity, $E_0 = peak electric field$, c = speed of light, epsilo$n_0 = permittivity$ W/m^2
Time-averaged over a full cycle
Graphs and experiments
Orthogonal Sinusoidal Oscillations of E and B
Draw: A 3D coordinate system showing propagation along the z-axis, electric field oscillations in the x-z plane (sine wave), and magnetic field oscillations in the y-z plane (cosine wave, phase-locked with E).
Interpret: Visually demonstrates that E and B are mutually perpendicular at every point and both are perpendicular to the direction of energy transport (z-axis).
Exam use: Frequently asked in TGT/PGT multiple-choice questions to identify field orientations when wave propagation direction is given.
Malus Law / Polarization Verification Experiment
Draw: A source of unpolarized light passing through a polarizer (P1) and an analyzer (P2) rotated at angle \theta, showing transmitted intensity variation.
Interpret: Confirms the transverse nature of waves because longitudinal waves would show no extinction when the analyzer is rotated.
Exam use: Tested via conceptual questions on why longitudinal waves (like sound waves in fluids) cannot be polarized.
Hertz's Electromagnetic Wave Experiment
Draw: An induction coil connected to spark gap transmitter spheres and a resonant wire loop receiver with a microscopic spark gap.
Interpret: Proved the existence of electromagnetic waves and showed spark generation depends on the orientation of the receiving loop relative to the electric field vector.
Exam use: Historical and conceptual context for transverse wave detection and spark resonance.
Solved numericals
TGT core
The magnetic field amplitude in an electromagnetic wave is $B_0 = 3 \times 10^{-8} T$. What is the amplitude of the electric field E_0?
Recall the relation between electric and magnetic field amplitudes: $E_0 = c B_0$.
Substitute the speed of light $c = 3 \times 10^8 m/s$ and $B_0 = 3 \times 10^{-8} T$.
A plane electromagnetic wave travels in vacuum along the x-direction. If the electric field is given by $E_y = 100 \cos(10^7 t - kx) V/m$, calculate the propagation constant k, the frequency f, and the expression for the associated magnetic field B_z.
Identify the angular frequency omega from the given phase: omeg$a = 10^7 rad/s$.
Find magnetic field amplitude $B_0 = E_0 / c = 100 / (3 \times 10^8) = 3.33 \times 10^{-7} T$.
Write the magnetic field vector equation respecting the right-hand rule (E x B points along +x): B_z(x,t) = 3.33 \times 10^{-7} cos(10^7 t - 0.033 x) Tesla.
Find the average intensity (energy per unit area per unit time) of an electromagnetic wave in free space if the peak value of the electric field is 120 V/m. Given epsilo$n_0 = 8.85 \times 10^{-12} C^2/N \cdot m^2$ and $c = 3 \times 10^8 m/s$.
Use the intensity formula for electromagnetic waves: $I = (1/2) c \epsilon_0 E_0^2$.
Substitute the given values: $c = 3 \times 10^8$, epsilo$n_0 = 8.85 \times 10^{-12}$, $E_0 = 120$.
Assumes propagation in ideal isotropic, homogeneous, charge-free, and current-free free space unless stated otherwise.
Derivations of transverse nature assume plane wave approximations where wavefronts are infinite planes perpendicular to the propagation vector.
Common misconceptions
Common error: Electromagnetic waves have longitudinal electric field components along the direction of travel.
Correct: Electromagnetic waves in free space are purely transverse; both E and B vectors are strictly perpendicular to the direction of propagation.
Maxwell's divergence equations in free space (div E = 0 and div B = 0) mathematically eliminate any longitudinal field components.
Common error: The electric field and magnetic field in an electromagnetic wave are out of phase.
Correct: E and B oscillate in phase with each other in a propagating plane electromagnetic wave.
The spatial and temporal derivatives in Maxwell's curl equations link them such that maximum E corresponds directly to maximum B at the same point in space and time.
Quick revision
E and B fields are mutually perpendicular and both are perpendicular to wave velocity c (E x B points along c).
Sound waves are longitudinal and cannot be polarized, whereas light and EM waves are transverse and polarizable.
Must remember
The electric field vector $\mathbf{E}$, magnetic field vector $\mathbf{B}$, and propagation vector $\mathbf{k}$ form a mutually orthogonal triad, with $\mathbf{E} \times \mathbf{B}$ pointing in the direction of wave velocity.
Electromagnetic waves in free space are strictly transverse; divergence equations $\nabla \cdot \mathbf{E} = 0$ and $\nabla \cdot \mathbf{B} = 0$ rule out any longitudinal components.
The magnitudes of electric and magnetic fields are related by $E_0 = c B_0$, where $c$ is the speed of light in vacuum.
The characteristic impedance of free space is given by $Z_0 = \sqrt{\frac{\mu_0}{\epsilon_0}} \approx 376.7 \omega$.
Electric and magnetic fields in a plane electromagnetic wave oscillate completely in phase with each other.
Glossary
Poynting Vector ($mathbf{S}$)The directional energy flux density vector of an electromagnetic field, given by $mathbf{S} = \frac{1}{mu_0}(mathbf{E} \times mathbf{B})$.
Characteristic Impedance of Free Space ($Z_0$)The ratio of the electric field strength to the magnetic field strength in a free-space electromagnetic wave, equal to $sqrt{\frac{mu_0}{epsilon_0}}$.
Propagation Vector ($mathbf{k}$)A vector whose magnitude $k = \frac{2pi}{lambda}$ indicates the wave number and whose direction specifies the line of wave propagation.
PolarizationA property applying to transverse waves specifying the geometrical orientation of the oscillating electric field vector.
Electric Field vs Magnetic Field in EM Waves
Parameter
Electric Field ($mathbf{E}$)
Magnetic Field ($mathbf{B}$)
Amplitude Relation
$E_0$
$B_0 = E_0 / c$
Unit
V/m
Tesla (T)
Oscillation Phase
In phase with $\mathbf{B}$
In phase with $\mathbf{E}$
Role in Force
Exerts primary electrostatic/magnetic force on charges
Exerts magnetic force proportional to velocity $\mathbf{v} \times \mathbf{B}$
Practice Quiz
Choose an option to reveal the explanation.
1. Which of the following vectors is parallel to the direction of energy propagation of an electromagnetic wave?
Explanation: The Poynting vector $\mathbf{S}$, which gives the direction and magnitude of energy flux, is proportional to $\mathbf{E} \times \mathbf{B}$.
2. What is the ratio of the magnitude of the electric field to the magnetic field ($E/B$) in a plane electromagnetic wave traveling in vacuum?
Explanation: Since $E_0 = c B_0$, the ratio $E/B$ equals the speed of light in vacuum, $c$.
3. The characteristic impedance of free space $Z_0$ is given by which of the following expressions?
Explanation: The characteristic impedance of free space is defined as $Z_0 = \sqrt{\frac{\mu_0}{\epsilon_0}}$, with a numerical value of approximately $376.7 \omega$.
4. Why do electromagnetic waves not have a longitudinal component in free space?
Explanation: In charge-free and current-free space, $\nabla \cdot \mathbf{E} = 0$ and $\nabla \cdot \mathbf{B} = 0$ enforce that no field components can oscillate along the direction of propagation $\mathbf{k}$.
5. What is the phase difference between the electric field $\mathbf{E}$ and magnetic field $\mathbf{B}$ oscillations in a plane electromagnetic wave propagating in vacuum?
Explanation: Maxwell's curl equations couple $\mathbf{E}$ and $\mathbf{B}$ such that they reach their maxima and minima simultaneously, meaning they are completely in phase ($0$ phase difference).
6. If an electromagnetic wave propagates along the positive z-axis, which axes can the electric and magnetic fields oscillate along?
Explanation: Propagation along z implies both $\mathbf{E}$ and $\mathbf{B}$ must lie in the xy-plane and be perpendicular to each other.
7. What is the unit of the Poynting vector $\mathbf{S}$?
Explanation: The Poynting vector represents energy flux density, which is power per unit area, measured in Watts per square meter ($\text{W/m}^2$).
8. Which of the following equations correctly expresses the speed of light $c$ in terms of free space permittivity $\epsilon_0$ and permeability $\mu_0$?
Explanation: From Maxwell's wave equation derivation, the speed of electromagnetic waves in vacuum is $c = \frac{1}{\sqrt{\mu_0 \epsilon_0}}$.
9. The phenomenon of polarization clearly demonstrates which characteristic of light waves?
Explanation: Only transverse waves can be polarized, as their oscillations can be restricted to a specific plane perpendicular to the direction of propagation.
10. What is the time-averaged intensity $I$ of a plane electromagnetic wave with peak electric field $E_0$ in vacuum?
Explanation: The time-averaged intensity of an electromagnetic wave is given by $I = \frac{1}{2} c \epsilon_0 E_0^2$.
11. Which Maxwell equation directly leads to the relation forcing the electric field to be perpendicular to the wave vector $\mathbf{k}$?
Explanation: Faraday's law of induction ($\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$) yields $\mathbf{k} \times \mathbf{E} = \omega \mathbf{B}$, proving $\mathbf{E}$ is perpendicular to $\mathbf{k}$.
12. If the peak value of the magnetic field in a plane electromagnetic wave is $2 \times 10^{-6} \text{T}$, what is the peak value of the electric field? (Take $c = 3 \times 10^8 \text{m/s}$)
Explanation: Using $E_0 = c B_0$, we get $E_0 = (3 \times 10^8 \text{m/s}) \times (2 \times 10^{-6} \text{T}) = 600 \text{V/m} = 6 \times 10^2 \text{V/m}$.
13. What happens when unpolarized light passes through a pair of ideal polarizers whose transmission axes are adjusted to be perpendicular to each other (crossed polaroids)?
Explanation: Crossed polaroids block all transmitted light (extinction), confirming the transverse vector model of electromagnetic waves.
14. In a harmonic electromagnetic wave expressed as $\mathbf{E}(z,t) = E_0 \cos(kz - \omega t)hat{i}$, what does the term $kz - \omega t$ represent?
Explanation: The argument of the cosine function, $kz - \omega t$, represents the phase of the harmonic wave at position $z$ and time $t$.
15. Which of the following relations correctly connects angular frequency $\omega$, wave number $k$, and wave speed $c$ in free space?
Explanation: The phase velocity is given by $v = \frac{\omega}{k} = c$, which rearranges to $\omega = c k$.
PYQ Patterns
These are pattern-based practice questions. Authentic year labels are only used when verified from a source.
1. The electric and magnetic fields in an electromagnetic wave are given by $E$ and $B$. The direction of propagation of the electromagnetic wave is along:
Explanation: The energy and wavefront propagation of an EM wave travels in the direction defined by the Poynting vector, which points along $\mathbf{E} \times \mathbf{B}$.
2. The dimensions of $\frac{1}{\sqrt{\mu_0 \epsilon_0}}$ are the same as:
Explanation: The expression $\frac{1}{\sqrt{\mu_0 \epsilon_0}}$ represents the speed of light $c$, whose dimensional formula is $[\mathrm{L}\mathrm{T}^{-1}]$, matching velocity.
3. If the amplitude of the magnetic field in a vacuum electromagnetic wave is $B_0$, what is the peak energy density associated with the magnetic field?
Explanation: The instantaneous magnetic energy density is $u_B = \frac{B^2}{2mu_0}$, so its peak value for a sinusoidal wave is $\frac{1}{2} \frac{B_0^2}{\mu_0}$.
4. Which of the following statements is correct regarding electromagnetic waves?
Explanation: In an electromagnetic wave, $\mathbf{E}$ and $\mathbf{B}$ are mutually perpendicular to each other and to the direction of propagation, and they oscillate completely in phase.
5. In free space, a plane electromagnetic wave has a peak electric field of $30 \text{V/m}$. The peak magnetic field is approximately:
6. The property of polarization exhibited by light waves proves that light is:
Explanation: Polarization is exclusively a characteristic of transverse waves, confirming that light waves are transverse in nature.
Timed Topic Test
Attempt these questions under a ten-minute limit using a separate timer.
1. An electromagnetic wave travels in free space with electric field given by $\mathbf{E} = E_0 \sin(kz - \omega t)hat{i}$. The corresponding magnetic field $\mathbf{B}$ is:
Explanation: Since $\mathbf{E} \times \mathbf{B}$ must point along $+hat{k}$ (the direction of propagation $z$), $(hat{i} \times hat{j} = hat{k})$, the magnetic field must oscillate along the y-axis with amplitude $B_0 = E_0/c$ and in phase.
2. What is the numerical value of the characteristic impedance of free space $Z_0$ approximately equal to?
3. If the electric field amplitude of a plane electromagnetic wave is $E_0$ and magnetic field amplitude is $B_0$, what is the total instantaneous energy density $u$ stored in the wave?
Explanation: Total energy density in an EM field is the sum of electric energy density ($u_E = \frac{1}{2}\epsilon_0 E^2$) and magnetic energy density ($u_B = \frac{1}{2mu_0} B^2$).
4. Which Maxwell equation leads to the deduction that magnetic field lines have no sources or sinks (non-existence of magnetic monopoles), forcing $\nabla \cdot \mathbf{B} = 0$?
Explanation: Gauss's law for magnetism states that $\nabla \cdot \mathbf{B} = 0$, reflecting that magnetic field lines are continuous loops.
5. In a region of space where $\mathbf{E}$ and $\mathbf{B}$ fields of an EM wave are present, the ratio of the energy density stored in the electric field to that stored in the magnetic field is:
Explanation: Since $E = cB$ and $c = \frac{1}{\sqrt{\mu_0 \epsilon_0}}$, the electric energy density $\frac{1}{2}\epsilon_0 E^2$ equals the magnetic energy density $\frac{1}{2mu_0} B^2$, giving a 1:1 ratio.
6. A laser beam has an intensity $I$. The RMS value of the electric field $E_{\mathrm{rms}}$ in the beam is related to $I$ by:
Explanation: Since $E_{\mathrm{rms}} = \frac{E_0}{\sqrt{2}}$, the average intensity $I = \frac{1}{2} c \epsilon_0 E_0^2$ becomes $I = c \epsilon_0 E_{\mathrm{rms}}^2$.
7. Which of the following field configurations is impossible for a plane electromagnetic wave traveling in the positive x-direction?
Explanation: If $E$ is along the x-direction (direction of propagation), it violates the transverse condition. Thus, $E$ along x and $B$ along y is impossible.
8. The angular frequency of an electromagnetic wave is $2pi \times 10^8 \text{rad/s}$. What is its propagation constant (wave number) $k$ in free space?
10. If the electric field amplitude of a plane wave is doubled, what happens to the average intensity of the electromagnetic wave?
Explanation: Intensity is proportional to the square of the electric field amplitude ($I \propto E_0^2$). Doubling $E_0$ increases the intensity by a factor of $2^2 = 4$.