13.4 Maxwell’s Equations

• Gauss's Law for Electricity is useful for calculating electric fields of symmetric charge distributions.


  True
• Why is the total magnetic flux through a closed surface always zero according to Gauss's Law for Magnetism?

  Magnetic fields are divergence-free
• The mathematical expression for Gauss's Law for Electricity is $\oint \vec{E} \cdot d\vec{A} =$$\frac{Q_{\text{enc}}}{\epsilon_{0}}$


  True
• Gauss's Law simplifies electric field calculations for symmetric charge distributions.

  True
• For a point charge enclosed by a spherical Gaussian surface, the electric field is constant at any point on the surface.


  True
• Gauss's Law for Magnetism states that the total magnetic flux through any closed surface is zero.

  True
• The unit of the magnetic field vector $\vec{B}$ is Tesla
• Gauss's Law for Magnetism states that the total magnetic flux through any closed surface is always zero
• Faraday's Law of Induction states that the induced electromotive force (emf) in a closed loop is proportional to the rate of change of magnetic flux
• Ampère-Maxwell's Law states that the line integral of the magnetic field around a closed loop is proportional to the electric current plus the rate of change of electric flux
• The term $\epsilon_{0} \frac{d\Phi_{E}}{dt}$ in Ampère-Maxwell's Law represents the displacement current
• The mathematical expression for Gauss's Law for Electricity is \oint \vec{E} \cdot d\vec{A} = \frac{Q_{\text{enc}}}{\epsilon_0}
• Gauss's Law for Magnetism states that the total magnetic flux through any closed surface is always zero
• Gauss's Law for Electricity states that the total electric flux through a closed surface is proportional to the electric charge enclosed
• The value of the vacuum permittivity $\epsilon_{0}$ is approximately $8.85 \times 10^{ - 12}$
• Match the symbols with their units:

  $\vec{E}$ ↔️ N/C

  $d\vec{A}$ ↔️ m²

  $Q_{\text{enc}}$ ↔️ C

  $\epsilon_{0}$ ↔️ C²/N·m²
• Steps to calculate electric field using Gauss's Law for symmetric charge distributions:

  1️⃣ Choose a suitable Gaussian surface

  2️⃣ Calculate the total charge enclosed by the surface

  3️⃣ Determine the electric flux through the surface

  4️⃣ Apply Gauss's Law to find the electric field
• Gauss's Law for Magnetism reflects the fact that magnetic fields are divergence-free.


  True
• The total magnetic flux through a closed surface enclosing a horseshoe magnet is zero.


  True
• Magnetic fields are divergence-free, meaning there are no magnetic charges or monopoles.

  True
• The negative sign in Faraday's Law indicates that the induced emf opposes the change in magnetic flux, as described by Lenz's Law.


  True
• The vacuum permeability $\mu_{0}$ has a value of 4\pi \times 10^{ - 7} \, \text{N} / \text{A}^{2}</latex>.


  True
• What does Gauss's Law for Electricity state about the total electric flux through a closed surface?

  Proportional to enclosed charge
• What is the value of the vacuum permittivity, $\epsilon_{0}$?


  $8.85 \times 10^{ - 12} \, \text{C}^{2} / \text{N}\cdot\text{m}^{2}$
• Gauss's Law for Magnetism reflects the absence of magnetic monopoles.


  True
• Match the symbols with their descriptions:

  $\vec{E}$ ↔️ Electric field vector

  $d\vec{A}$ ↔️ Differential area vector

  $Q_{\text{enc}}$ ↔️ Total charge enclosed

  $\epsilon_{0}$ ↔️ Vacuum permittivity
• The unit of the electric field vector $\vec{E}$ is N/C
• The integral $\oint \vec{E} \cdot d\vec{A}$ for a point charge enclosed by a spherical Gaussian surface simplifies to $\frac{q}{\epsilon_{0}}$
• The mathematical expression for Gauss's Law for Magnetism is \oint \vec{B} \cdot d\vec{A} = 0</latex>
• Magnetic flux forms closed loops
• What is the mathematical expression for Gauss's Law for Magnetism?

  $\oint \vec{B} \cdot d\vec{A} =$$0$
• What is the mathematical expression for Faraday's Law of Induction?

  \varepsilon = - \frac{d\Phi_B}{dt}</latex>
• What is the induced emf in a coil with 100 turns when the magnetic flux changes from 0.2 T to 0.5 T in 2 seconds over an area of 0.1 m²?

  $- 1.5 \, \text{V}$
• What is the mathematical expression for Ampère-Maxwell's Law?

  $\oint \vec{B} \cdot d\vec{l} =$$\mu_{0} \left( I + \epsilon_{0} \frac{d\Phi_{E}}{dt} \right)$