Electromagnetism and Charge Conservation Practice Quiz (12 questions)

All questions and answers

1. A field measurement gives div⁡(E⃗)=ρ(M,t)ε0\operatorname{div}(\vec E)=\frac{\rho(M,t)}{\varepsilon_0} and a time-varying magnetic field. Which statements correctly identify the local electric-field equations?

Maxwell–Gauss relates electric-field divergence to charge density divided by vacuum permittivity.
Maxwell–Gauss relates electric-field divergence to current density divided by vacuum permittivity.
Maxwell–Faraday relates electric-field curl to the negative time derivative of magnetic field.
Maxwell–Faraday relates electric-field divergence to the negative time derivative of magnetic field.
Maxwell–Gauss relates electric-field curl to charge density divided by vacuum permittivity.

Maxwell–Gauss relates electric-field divergence to charge density divided by vacuum permittivity. · Maxwell–Faraday relates electric-field curl to the negative time derivative of magnetic field.

Explanation

Maxwell–Gauss gives div⁡(E⃗)=ρ/ε0\operatorname{div}(\vec E)=\rho/\varepsilon_0, while Maxwell–Faraday gives rot⁡(E⃗)=−∂B⃗/∂t\operatorname{rot}(\vec E)=-\partial\vec B/\partial t. The false statements swap divergence with curl or replace charge density with current density.

2. In a region, the electric field changes with time and a current density is present. Which statements correctly describe the local magnetic-field equations?

The displacement term in Maxwell–Ampère is proportional to the electric field’s time derivative.
The local magnetic-flux equation sets the magnetic-field curl equal to zero.
The magnetic-field curl includes the magnetic-field time derivative multiplied by vacuum permeability.
The magnetic-field curl includes the current-density term multiplied by vacuum permeability.
The magnetic-field divergence is zero, expressing the local magnetic-flux equation.

The displacement term in Maxwell–Ampère is proportional to the electric field’s time derivative. · The magnetic-field curl includes the current-density term multiplied by vacuum permeability. · The magnetic-field divergence is zero, expressing the local magnetic-flux equation.

Explanation

Maxwell–Ampère’s curl equation contains the current-density term multiplied by μ0\mu_0 and the displacement term proportional to the time derivative of the electric field. The magnetic-flux equation states that the magnetic-field divergence is zero; this does not require its curl to be zero. Charge density and the magnetic-field time derivative do not provide the terms stated for the magnetic-field curl.

3. A surface carries a nonuniform charge density σ(M)\sigma(M). Which statements correctly describe its density and total charge?

The total surface charge is obtained by integrating density over the charged surface.
At a point, surface charge density is the local charge element divided by a length element.
At a point, surface charge density is the local charge element divided by its area element.
For a small patch, the charge element equals density multiplied by its area element.
The total surface charge is obtained by integrating density along the surface boundary.

The total surface charge is obtained by integrating density over the charged surface. · At a point, surface charge density is the local charge element divided by its area element. · For a small patch, the charge element equals density multiplied by its area element.

Explanation

Surface density is σ(M)=dq(M)/dS(M)\sigma(M)=dq(M)/dS(M), and the total charge is QS=∬Sσ(M) dSQ_S=\iint_S\sigma(M)\,dS. Thus a patch carries dq=σ dSdq=\sigma\,dS; integrating along a boundary or dividing by a length instead describes the wrong measure.

4. A surface is crossed by a mixture of carrier types whose densities, charges, and velocities may differ. Which statements describe the total current density and surface current?

The surface current is obtained by integrating current density over the enclosed volume.
The total current through a surface is the surface integral of the magnitude of current density.
A carrier’s contribution to total current density is its charge divided by density, multiplied by velocity.
The total current density combines the contributions from the different carrier types by summation.
Each carrier type contributes through its number density, charge, and velocity vector.

The total current density combines the contributions from the different carrier types by summation. · Each carrier type contributes through its number density, charge, and velocity vector.

Explanation

For multiple carrier types, total current density is the sum of each type’s number density–charge–velocity contribution. Current through a surface is instead the integral of the current-density vector dotted with the oriented surface element; it is not a magnitude integral or a volume integral.

5. Charge density and current vary in space and time in a conducting region. Which statements correctly express local charge conservation and the charge in a fixed volume?

In three dimensions, local conservation combines the time derivative of charge density with the divergence of current density.
The charge inside a fixed volume is obtained by integrating charge density over that volume.
In one dimension, charge conservation relates the time derivative of density to the spatial derivative of current.
For a fixed volume, the enclosed charge may depend on time through the density distribution.
The one-dimensional conservation equation adds the time derivative of density to the time derivative of current.

In three dimensions, local conservation combines the time derivative of charge density with the divergence of current density. · The charge inside a fixed volume is obtained by integrating charge density over that volume. · In one dimension, charge conservation relates the time derivative of density to the spatial derivative of current. · For a fixed volume, the enclosed charge may depend on time through the density distribution.

Explanation

The one-dimensional equation uses the time derivative of charge density and the spatial derivative of current, while the three-dimensional equation uses the divergence of current density. Enclosed charge is the volume integral of charge density and can vary with time; the one-dimensional current term is a spatial, not temporal, derivative.

6. A fixed closed volume is crossed by a current density field. Which statements correctly describe the outward-current convention and the surface-current integral?

An inward-directed current density corresponds to charge entering the enclosed volume.
When current density points outward, charge leaves the volume and its contained charge decreases.
The surface-current integral uses the inward area element as its positive orientation.
The surface integral accounts for contributions across the entire closed boundary, not just one patch.
The signed current through the closed surface is the integral of current density dotted with the outward area element.

An inward-directed current density corresponds to charge entering the enclosed volume. · When current density points outward, charge leaves the volume and its contained charge decreases. · The surface integral accounts for contributions across the entire closed boundary, not just one patch. · The signed current through the closed surface is the integral of current density dotted with the outward area element.

Explanation

The signed current is the closed-surface integral of current density dotted with the outward element, and outward flow removes charge while inward flow brings charge in. The integral covers the whole boundary; using inward orientation as the positive convention reverses the stated outward convention.

7. A conductor is in a stationary regime. Which statements correctly characterize its charge density and current through a closed surface?

The local time derivative of charge density vanishes in the stationary regime.
The net current through a closed surface may be nonzero in a stationary regime.
The divergence of current density is zero at each point in the stationary regime.
A nonzero divergence of current density is compatible with stationarity when charge density is fixed.
The general continuity equation requires charge density to remain constant in every regime.

The local time derivative of charge density vanishes in the stationary regime. · The divergence of current density is zero at each point in the stationary regime.

Explanation

Stationarity gives zero time variation of charge density, and the continuity equation then gives zero divergence of current density. The general continuity equation permits time-varying density, while a stationary regime has zero total current through any closed surface.

8. A closed surface encloses charge, while an electric field changes with time across a surface bounded by a contour. Which statements correctly apply the integral laws?

The electric-field flux through a closed surface equals the enclosed charge multiplied by vacuum permittivity.
Electric charges act as sources of the electric field E⃗\vec E.
The outward electric-field flux through the closed surface equals enclosed charge divided by vacuum permittivity.
The generalized Ampère law replaces the electric-field variation term with a magnetic-field variation term.
The generalized Ampère law includes a term involving the time derivative of the electric field.

Electric charges act as sources of the electric field $$\vec E$$. · The outward electric-field flux through the closed surface equals enclosed charge divided by vacuum permittivity. · The generalized Ampère law includes a term involving the time derivative of the electric field.

Explanation

Gauss’s law gives electric flux as enclosed charge divided by vacuum permittivity, and electric charges are sources of E⃗\vec E. Generalized Ampère’s law includes the changing-electric-field term; multiplying by permittivity or substituting a magnetic-field variation gives the wrong relation.

9. A uniform magnetic field crosses a flat surface at an angle. Which statements describe the flux and the induced circulation for an immobile surface?

The electric-field circulation around the boundary equals the negative time derivative of magnetic flux.
The magnetic flux is obtained by integrating the field’s tangential component over the surface.
The electric-field circulation equals the positive time derivative of electric flux.
The magnetic flux is obtained by integrating the field’s normal component over the surface.
The magnetic flux through the surface is defined as the line integral of the electric field.

The electric-field circulation around the boundary equals the negative time derivative of magnetic flux. · The magnetic flux is obtained by integrating the field’s normal component over the surface.

Explanation

Flux integrates the magnetic field’s component along the oriented surface element. For an immobile surface, Maxwell–Faraday gives circulation equal to the negative time derivative of magnetic flux. Integrating the tangential component over the boundary does not give magnetic flux, and the positive derivative of electric flux is not the stated circulation law.

10. What role do the four coupled equations in the electric and magnetic fields serve?

They are useful for describing electrostatic equilibrium alone.
They are used to describe only the magnetic field of steady currents.
They describe gravitational waves through coupled electric and magnetic fields.
They characterize the propagation of sound waves through a medium.
They are useful for describing electromagnetic waves.

They are useful for describing electromagnetic waves.

Explanation

The four coupled equations in E⃗\vec E and B⃗\vec B are useful for describing electromagnetic waves. Electrostatic equilibrium alone, steady-current magnetic fields alone, sound waves, and gravitational waves do not match that stated role.

11. Across a charged surface with surface charge density σ\sigma, which statements give the field relation at the two labeled points?

The field sum E⃗(P1)+E⃗(P2)\vec E(P_1)+\vec E(P_2) equals σε0\frac{\sigma}{\varepsilon_0}.
The magnitude of the specified field difference is set by ∣σ∣ε0\frac{|\sigma|}{\varepsilon_0}.
For positive surface charge, the specified field difference points along the positive normal.
The field difference E⃗(P1)−E⃗(P2)\vec E(P_1)-\vec E(P_2) equals σε0\frac{\sigma}{\varepsilon_0}.
The specified field difference equals ε0σ\frac{\varepsilon_0}{\sigma}.

The magnitude of the specified field difference is set by $$\frac{|\sigma|}{\varepsilon_0}$$. · For positive surface charge, the specified field difference points along the positive normal. · The field difference $$\vec E(P_1)-\vec E(P_2)$$ equals $$\frac{\sigma}{\varepsilon_0}$$.

Explanation

The stated relation is E⃗(P1)−E⃗(P2)=σε0\vec E(P_1)-\vec E(P_2)=\frac{\sigma}{\varepsilon_0}, so positive charge gives a positive signed difference and its magnitude is ∣σ∣/ε0|\sigma|/\varepsilon_0. The reciprocal expression and the field sum do not follow from the relation.

12. For a surface-charge density of magnitude 2ε0E02\varepsilon_0 E_0, which conclusions follow for the labeled field values?

The magnitude of E⃗(P1)−E⃗(P2)\vec E(P_1)-\vec E(P_2) is 2E02E_0.
For positive charge density, the signed difference E⃗(P1)−E⃗(P2)\vec E(P_1)-\vec E(P_2) is positive along the stated normal.
The magnitude of the field difference is E0/2E_0/2.
The relation requires E⃗(P1)+E⃗(P2)=2E0\vec E(P_1)+\vec E(P_2)=2E_0.
The field difference is independent of surface-charge density.

The magnitude of $$\vec E(P_1)-\vec E(P_2)$$ is $$2E_0$$. · For positive charge density, the signed difference $$\vec E(P_1)-\vec E(P_2)$$ is positive along the stated normal.

Explanation

Substitution into the surface relation gives E⃗(P1)−E⃗(P2)=σ/ε0\vec E(P_1)-\vec E(P_2)=\sigma/\varepsilon_0, whose magnitude is 2E02E_0; positive charge gives a positive signed difference in the stated orientation. The half-sized value, field-sum claim, and density independence contradict the given relation.

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Lock in the answers with 37 flashcards on Electromagnetism and Charge Conservation.

What is Maxwell-Gauss’ local equation?

div⁡(E⃗)=ρ(M,t)ε0\operatorname{div}(\vec E)=\frac{\rho(M,t)}{\varepsilon_0}

What is Maxwell-Faraday’s local equation?

rot⁡(E⃗)=−∂B⃗∂t\operatorname{rot}(\vec E)=-\frac{\partial \vec B}{\partial t}

What is the local magnetic-flux equation?

div⁡(B⃗)=0\operatorname{div}(\vec B)=0

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