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Electricity and Magnetism
Displacement Current
Displacement current is a quantity appearing in Maxwell's equations that is defined in terms of the rate of change of electric displacement field or electric flux. It was introduced by James Clerk Maxwell to resolve the inconsistency in Ampere's Circuital Law for non-steady currents.
UP TGT ScienceUP PGT PhysicsStudy Notes + Practice
Scope and prerequisites
TGT core
Definition of displacement current, relation with changing electric flux, identification in charging capacitors, and modification of Ampere's law.
PGT extension
Derivation of Maxwell's modified curl equation for magnetic field, symmetry of Maxwell's equations, magnetic field between capacitor plates, and wave equation for electromagnetic waves in free space.
Prerequisites
Gauss's Law in electrostatics
Ampere's Circuital Law
Time-varying electric and magnetic fields
Failure of Original Ampere's Law
Ampere's Circuital Law states that the line integral of magnetic field B along a closed loop is equal to μ0 times the conduction current passing through the surface.
Consider a charging parallel-plate capacitor: applying Ampere's law to two different surfaces bounded by the same loop yields contradictory results.
Surface S1 intersects the wire (passing conduction current i), while surface S2 passes between the capacitor plates (where conduction current is zero).
Introduction of Displacement Current
To resolve the discontinuity, Maxwell postulated the existence of an additional current due to changing electric flux, termed displacement current id.
It is mathematically expressed as id = ε0 (dΦE/dt).
Displacement current does not involve actual movement of charge carriers; rather, it is associated with the time-varying electric field or polarization in a dielectric.
Modified Ampere-Maxwell Law
The total current enclosed by a loop is the sum of conduction current (ic) and displacement current (id).
The modified Ampere's law in integral form is ∮ B · d$l = \mu_{0} (i_{c} + \epsilon_{0} d\Phi_{E}/dt)$.
In differential form, it is written as ∇ × $B = \mu_{0} J + \mu_{0}\epsilon_{0} (\partial E / \partial t)$.
Physical Significance and Continuity
Displacement current ensures the continuity of current through a capacitor; when conduction current drops to zero inside the plates, displacement current takes its exact value.
It is the source of magnetic fields in regions devoid of free charges or conduction currents, such as in empty space for propagating electromagnetic waves.
The product μ0ε0 is inversely proportional to the square of the speed of light in vacuum (c2).
Derivations
Derivation of Displacement Current in a Charging Capacitor
Conditions: Parallel-plate capacitor connected to a time-varying current source
Let the charge on the capacitor plates at any instant t be q, and the area of each plate be A.
The electric field E between the plates is given by $E = q / (\epsilon_{0} A)$.
The electric flux ΦE through the surface between the plates is ΦE = E · $A = q / \epsilon_{0}$.
Differentiating both sides with respect to time t gives dΦE/d$t = (1 / \epsilon_{0}) (dq/dt)$.
Since dq/d$t = i_{c} (conduction current)$, we obtain ε0 (dΦE/dt) = ic, which defines the magnitude of the displacement current id.
∇ × B = curl of magnetic field, J = conduction current density N / (A·m2)
Differential point form
Magnetic Field between Circular Capacitor Plates
$$B = \frac{\mu_0 i r}{2 \pi R^2}$$
$r = radial distance from axis (r \langle R)$, R = radius of plates, i = charging current Tesla (T)
At radius r inside circular parallel-plate capacitor
Electric Flux for Parallel-Plate Capacitor
$$\phi_E = \frac{q}{\epsilon_0}$$
ΦE = electric flux, q = charge on plate, ε0 = permittivity V·m
Uniform electric field approximation
Speed of Electromagnetic Waves
$$c = \frac{1}{\sqrt{\mu_0 \epsilon_0}}$$
c = speed of light in vacuum, μ0 = permeability, ε0 = permittivity m / s
Free space
Total Enclosed Current
$$I_{total} = I_c + I_d$$
Itotal = total effective current producing magnetic field Ampere (A)
Any arbitrary surface bounded by a closed loop
Graphs and experiments
Magnetic Field vs. Radial Distance in a Charging Capacitor
Draw: A linear increase of B with radius r for r < R, followed by an inverse proportionality (1/r) decrease for r > R.
Interpret: Inside the capacitor plates, displacement current is distributed uniformly across the area, causing B to grow linearly with r. Outside, all displacement/conduction current is effectively concentrated on the axis, leading to 1/r decay.
Exam use: Frequently tested in PGT numericals and conceptual graphs regarding field profiles.
Conduction vs. Displacement Current in an AC Circuit
Draw: A sinusoidal wave plot where conduction current in the connecting wires matches the displacement current across the capacitor gap in both amplitude and phase continuity.
Interpret: Proves that displacement current seamlessly continues conduction current across discontinuities like dielectric gaps or capacitors in alternating circuits.
Exam use: Conceptual verification questions in TGT examinations.
Maxwell's Symmetry Demonstration setup
Draw: Schematic diagram showing time-varying electric field generating magnetic field and vice versa, leading to self-sustaining electromagnetic wave propagation.
Interpret: Highlights the dual symmetry introduced by displacement current, completing Faraday's law of induction with Maxwell's correction.
Exam use: Short answer and assertion-reasoning questions in PGT physics.
Solved numericals
TGT core
A parallel-plate capacitor with circular plates of radius 5 cm is being charged at a constant rate such that the electric flux between the plates changes at the rate of 106 V·m/s. Calculate the displacement current.
Identify the given values: dΦE/d$t = 10^{6} V\cdot m/s$.
Recall the formula for displacement current: id = ε0 (dΦE/dt).
Substitute the value of permittivity of free space ε0 = 8.854 × 10-12 C2/N·m2.
Calculate: id = 8.854 × 10-12 × 106 = 8.854 × 10-6 A.
Answer: 8.85 μA
PGT extension
A parallel-plate capacitor having plate area 20 cm2 and separation 1.0 mm is connected to an AC source of voltage V = 100 sin(1000t) volts. Find the peak displacement current between the plates.
Calculate the capacitance $C = \epsilon_{0} A / d = (8.854 \times 10^{-12} \times 20 \times 10^{-4}) / (1 \times 10^{-3}) = 1.77 \times 10^{-11} F$.
The displacement current equals the conduction current in the circuit: id(t) = C (dV/dt).
The electric field in a region is given by $E = E_{0} \sin(\omegat - kx) j-cap$. Find the displacement current density vector Jd at that point.
Recall the formula for displacement current density: Jd = ε0 (&partial;E / &partial;t).
Perform partial differentiation of E with respect to time t: &partial;E / &partial;$t = \omega E_{0} \cos(\omegat - kx) j-cap$.
Multiply by ε0 to obtain the vector Jd.
Answer: epsilon_0 omega E_0 cos(omega t - kx) hat{j}
Assumptions and limitations
Assumes macroscopic electromagnetism where classical field approximations hold valid without quantum electrodynamic corrections.
Valid for isotropic and homogeneous dielectric media when relative permittivity εr is constant.
Common misconceptions
Common error: Displacement current involves the actual physical flow of free electrons across the capacitor gap.
Correct: Displacement current is due to the time-variation of electric flux or electric field, not moving charge.
No charge carriers move across the vacuum or dielectric barrier of a properly insulated capacitor.
Common error: Ampere's Circuital Law is completely incorrect and replaced by Maxwell's law.
Correct: Ampere's law is a special case valid only for steady conduction currents; Maxwell's modification makes it universally valid.
When electric fields do not change with time (dΦE/dt = 0), the displacement current term vanishes, reducing the equation back to original Ampere's law.
Quick revision
Displacement current id = ε0 (dΦE/dt) bridges the gap in Ampere's law for time-varying electric fields.
Displacement Current Definition: $i_d = \epsilon_0 \frac{dPhi_E}{dt}$
Displacement Current Density: $J_d = \epsilon_0 \frac{\partial E}{\partial t}$
Must remember
Displacement current ($i_d$) is given by $\epsilon_0 (d\Phi_E / dt)$ and arises due to a time-varying electric field.
The modified Ampere-Maxwell law in integral form is $\oint B \cdot dl = \mu_0 i_c + \mu_0 \epsilon_0 (d\Phi_E / dt)$.
Displacement current does not involve the physical transport or movement of free charge carriers.
The product $\mu_0 \epsilon_0$ is inversely proportional to the square of the speed of light in vacuum ($c^2$).
In differential form, the modified Ampere's law is expressed as $\nabla \times B = \mu_0 J + \mu_0 \epsilon_0 (\partial E / \partial t)$.
Glossary
Displacement CurrentAn equivalent current introduced by Maxwell, proportional to the time rate of change of electric flux, producing a magnetic field.
Conduction CurrentCurrent ($i_c$) resulting from the actual physical flow of free charge carriers or electrons.
Ampere-Maxwell LawThe generalized form of Ampere's Circuital Law that incorporates the displacement current term.
Displacement Current DensityVector quantity $J_d = \epsilon_0 (\partial E / \partial t)$ representing displacement current per unit area.
Conduction Current vs Displacement Current
Parameter
Conduction Current ($i_c$)
Displacement Current ($i_d$)
Origin
Flow of free charge carriers
Time-varying electric field or electric flux
Ohm's Law applicability
Applicable ($V = IR$)
Not applicable
Presence in Capacitor Gap
Zero
Non-zero (equals charging current)
Practice Quiz
Choose an option to reveal the explanation.
1. What is the mathematical expression for displacement current $i_d$?
Explanation: Displacement current is defined by Maxwell as $i_d = \epsilon_0 (d\Phi_E / dt)$.
2. Which of the following physical quantities is responsible for producing displacement current?
Explanation: Displacement current is generated when electric flux varies with time ($d\Phi_E / dt \neq 0$).
3. Does displacement current involve the actual movement of charge carriers?
Explanation: Displacement current does not involve real charge transport; it arises from time-varying electric fields or polarization.
4. What is the integral form of the modified Ampere-Maxwell Law?
Explanation: The modified Ampere's law accounts for both conduction and displacement currents: $\oint B \cdot dl = \mu_0 i_c + \mu_0 \epsilon_0 (d\Phi_E / dt)$.
5. In the differential form of the Ampere-Maxwell law, what term represents the displacement current density?
Explanation: The term $\mu_0 \epsilon_0 (\partial E / \partial t)$ corresponds to the displacement current density contribution in $\nabla \times B = \mu_0 J + \mu_0 \epsilon_0 (\partial E / \partial t)$.
6. What is the unit of displacement current?
Explanation: Being a current, displacement current is measured in Amperes (A).
7. If the electric flux through a surface is constant with time, what is the value of the displacement current?
Explanation: Since $i_d = \epsilon_0 (d\Phi_E / dt)$, a constant electric flux yields $d\Phi_E / dt = 0$, making $i_d = 0$.
8. How is the speed of light in vacuum $c$ related to $\mu_0$ and $\epsilon_0$?
Explanation: The speed of electromagnetic waves in free space is given by $c = 1 / \sqrt{\mu_0 \epsilon_0}$.
9. Why did original Ampere's Law fail for a charging parallel-plate capacitor?
Explanation: Surface $S_1$ enclosing conduction current gave $\mu_0 i$, while surface $S_2$ between plates gave zero, showing a mathematical contradiction.
10. What is the formula for displacement current density $J_d$?
Explanation: Displacement current density is $J_d = \epsilon_0 (\partial E / \partial t)$.
11. What is the magnitude of the magnetic field at a radial distance $r$ ($r < R$) inside a circular parallel-plate capacitor of radius $R$ carrying charging current $i$?
Explanation: Applying Ampere-Maxwell law inside the plates yields $B (2\pi r) = \mu_0 i (r^2 / R^2)$, leading to $B = \frac{\mu_0 i r}{2\pi R^2}$.
12. Which law was modified by Maxwell by adding the displacement current term?
Explanation: Maxwell added the displacement current term to Ampere's Circuital Law to make it universally consistent.
13. What is the unit of displacement current density $J_d$?
Explanation: Current density is current per unit area, measured in Amperes per square meter (A/m$^2$).
14. In regions devoid of free charges, what acts as the source of magnetic fields?
Explanation: In empty space or dielectric gaps (like in electromagnetic waves), time-varying electric fields create displacement currents which serve as magnetic field sources.
15. When does the modified Ampere-Maxwell law reduce exactly to the original Ampere's Circuital Law?
Explanation: Under steady-state conditions where $d\Phi_E / dt = 0$, the displacement current vanishes, reducing the equation to standard Ampere's Law.
PYQ Patterns
These are pattern-based practice questions. Authentic year labels are only used when verified from a source.
1. A parallel plate capacitor with circular plates of radius $R$ is being charged with a current $i$. What is the displacement current across the plates?
Explanation: To maintain continuity of current in a charging capacitor, the displacement current across the plates is exactly equal to the conduction current $i$ in the connecting wires.
2. The dimensions of the product $\mu_0 \epsilon_0$ are same as:
Explanation: Since $c = 1 / \sqrt{\mu_0 \epsilon_0}$, squaring both sides gives $c^2 = 1 / (\mu_0 \epsilon_0)$, so $\mu_0 \epsilon_0 = 1 / c^2$, which has dimensions of (Velocity)$^{-2}$.
3. If the electric field between the plates of a capacitor is changing at the rate of $10^6$ V/(m$\cdots) and plate area is $0.1$ m$^2$, what is the displacement current? (Take $\epsilon_0 = 8.85 \times 10^{-12}$ C$^2$/(N$\cdotm$^2$))
Explanation: Electric flux $\Phi_E = E \cdot A$, so $d\Phi_E/dt = A (dE/dt) = 0.1 \times 10^6 = 10^5$ V$\cdotm/s. Thus $i_d = \epsilon_0 (10^5) = 8.85 \times 10^{-12} \times 10^5 = 8.85 \times 10^{-7}$ A.
4. Which of the following Maxwell's equations explicitly contains the displacement current term?
Explanation: The curl of $B$ equation (Ampere-Maxwell law) includes the displacement current density term $\mu_0 \epsilon_0 (\partial E / \partial t)$.
5. A parallel plate capacitor is charged by a battery. Which of the following statements is true regarding the magnetic field?
Explanation: Conduction current in wires and displacement current between plates both act as sources of magnetic fields.
6. If the charge on a capacitor plate varies as $q = q_0 \sin(\omega t)$, what is the maximum displacement current?
Explanation: Displacement current $i_d = dq/dt = \frac{d}{dt}(q_0 \sin(\omega t)) = q_0 \omega \cos(\omega t)$. Its maximum value is $q_0 \omega$.
Timed Topic Test
Attempt these questions under a ten-minute limit using a separate timer.
1. The concept of displacement current was introduced primarily to satisfy:
Explanation: Displacement current resolves the mathematical discontinuity in Ampere's law, preserving charge conservation and current continuity.
2. What is the total effective current $I_{total}$ enclosed by an Amperian loop intersecting a charging capacitor?
Explanation: The total effective current producing the magnetic field is the sum of conduction current and displacement current: $I_{total} = I_c + I_d$.
3. In free space where there are no free charges ($J = 0$), what does the differential Ampere-Maxwell law become?
Explanation: With $J = 0$, the equation simplifies to $\nabla \times B = \mu_0 \epsilon_0 (\partial E / \partial t)$.
4. Calculate the displacement current when the electric flux changes at the rate of $0.05$ V$\cdotm/s. (Given $\epsilon_0 = 8.85 \times 10^{-12}$ SI units)
5. Which of the following statements about displacement current is incorrect?
Explanation: Displacement current does not involve actual charge flow through a resistance, hence it does not cause ohmic Joule heating ($I^2Rt$).
6. If a dielectric medium of permittivity $\epsilon = K \epsilon_0$ is placed between capacitor plates, how does the displacement current change compared to vacuum?
Explanation: Permittivity becomes $\epsilon = K \epsilon_0$, so displacement current $i_d = \epsilon (d\Phi_E / dt)$ increases by factor $K$.
7. What physical phenomenon is symmetrical to Faraday's law of electromagnetic induction (changing magnetic field producing electric field)?
Explanation: Faraday's law shows changing magnetic fields induce electric fields; displacement current demonstrates changing electric fields induce magnetic fields.
8. What is the dimension of displacement current density $J_d$?
Explanation: Current density is Current / Area, so its dimensional formula is $[A / L^2]$ or $[A L^{-2}]$. CUP/TGT level mechanics of units.
9. A capacitor is connected to an alternating voltage source $V = V_0 \sin(\omega t)$. The displacement current between the plates:
Explanation: For a capacitor, current leads voltage by $90^\circ$ because $i = C (dV/dt) \propto \cos(\omega t)$. Displacement current follows this same phase relationship.
10. Which scientist unified electricity, magnetism, and light by introducing the displacement current into electromagnetic theory?
Explanation: James Clerk Maxwell formulated the complete set of electromagnetic equations and predicted electromagnetic waves by introducing displacement current.