5.4: Electric Potential from Gauss Law

Learning Objectives

• Apply Gauss Law to charge distributions.
• Determine electric potential functions from electric fields using integrals.
• Use electric field and potential functions to determine charge density on a conductor.

Model

Image Credit: OpenStax University Physics Vol 2 (https://openstax.org/details/books/university-physics-volume-2)

Explore

1. What is the electric field inside either of the plates (they are conductors)?
2. How many faces of the Gaussian box have a non-zero electric flux? Shade those faces and draw a normal vector.
3. Compare the direction of \(\vec{E}\) and \(d\vec{A}\) (parallel, perpendicular, or neither).
4. Does the electric field magnitude vary with distance (for example, do the field lines converge or diverge as they leave the plate)?
5. Compare the direction of \(\vec{E}\) and \(d\vec{l}\) (parallel, perpendicular, or neither).

Critical Thinking & Concept Invention

1. Use Gauss's Law to identify the electric field magnitude created by the charge distribution.
   • Identify the enclosed charge.
   • Simplify the integral and explain how you were able to simplify it.
   • Identify the electric field in terms of \(\sigma\).
   • Identify the electric field in terms of capacitor charge \(Q\) if that charge is uniformly distributed over a plate with area \(A\).

   \[ Q_{\mathrm{enc}}= \]

   \[ \oint \vec{E}\cdot d\vec{A}=\frac{Q_{\mathrm{enc}}}{\varepsilon_0} \]

2. Determine the change in electric potential \((\Delta V)\) from the positive plate to the negative plate.
   • Simplify the integral and explain how you were able to simplify it.
   • Identify the electric potential in terms of \(\sigma\).
   • Identify the electric potential in terms of capacitor charge \(Q\) if that charge is uniformly distributed over a plate with area \(A\).
   • Using \(\Delta V_C=Q/C\), identify the capacitance \(C\).

   \[ \Delta V\equiv-\int\vec{E}\cdot d\vec{l} \]

   \[ \Delta V= \]

   \[ \Delta V= \]

   \[ C= \]

Summary

Electric potential and electric fields are related by the line integral:

\[ \Delta V\equiv-\int\vec{E}\cdot d\vec{l} \]

Gauss's Law relates electric field to charge on a conductor:

\[ \oint\vec{E}\cdot d\vec{A}=\frac{Q_{\mathrm{enc}}}{\varepsilon_0} \]

Combining these relationships allows electric potential to be related to charge. Since capacitance is defined as the ratio of charge to voltage:

\[ C\equiv\frac{Q}{\Delta V} \]

Apply

A cylindrical capacitor is created from a long thin wire (length \(L\)) and an outer conducting cylinder of the same length and radius \(R\). The inner wire has charge \(+Q\) and the outer cylinder has charge \(-Q\). Determine the capacitance.

Model

Image Credit: OpenStax University Physics Vol 2 (https://openstax.org/details/books/university-physics-volume-2)

Explore

1. What is the geometric shape of the charge distribution? What is the equation for its volume?
2. What is the shape of the Gaussian surface shown? What is the equation for its surface area?
3. Can Gauss's Law be reduced as shown below for this charge distribution and Gaussian surface? Explain why or why not.

\[ \Phi_E\equiv\iint\vec{E}\cdot d\vec{A}\longrightarrow \Phi_E=EA \]

Critical Thinking & Concept Invention

\[ \vec{E}= \]

1. Use Gauss's Law to identify the electric field magnitude created by the charged sphere if its total net charge is \(+Q\).

   \[ \oint\vec{E}\cdot d\vec{A}=\frac{Q_{\mathrm{enc}}}{\varepsilon_0} \]

2. What coordinate system (Cartesian, cylindrical, or spherical) best aligns with the direction of this electric field?
3. Identify a vector equation for the electric field by including an appropriate unit vector.
4. Determine the change in electric potential between:
   • The radius of the charged sphere and a point on the Gaussian surface.
   • A point at infinity and a point on the Gaussian surface.

   \[ \Delta V\equiv-\int\vec{E}\cdot d\vec{l} \]

   \[ \Delta V\equiv-\int\vec{E}\cdot d\vec{l} \]

5. Use the result referenced to infinity to identify the capacitance of the charged sphere.

   \[ \Delta V_C=\frac{Q}{C} \]

   \[ C= \]

Model

A sphere of radius \(R\) has a total net charge \(Q\) that is uniformly distributed throughout its volume. Two Gaussian surfaces are shown: one inside the sphere and one outside the sphere.

Image Credit: OpenStax University Physics Vol 2 (https://openstax.org/details/books/university-physics-volume-2)

Explore

1. For constant charge density, \(\rho=Q/V\), where \(V\) represents volume, what is the charge density of the sphere?
2. What is the volume enclosed by each Gaussian sphere?
3. For which Gaussian surface (inside or outside) is the entire charge \(Q\) enclosed?
4. For the surface that does not enclose all of \(Q\), what is the enclosed charge? Use the charge density relationship and geometric variables.

Critical Thinking & Concept Invention

1. Use Gauss's Law to identify the electric field magnitude created by the charge distribution.
   • Identify the enclosed charge in each case.
   • Simplify the integral and explain your reasoning.
   • Express the electric field in terms of \(Q\).

   \[ Q_{\mathrm{enc}}= \qquad Q_{\mathrm{enc}}= \]

   \[ \oint\vec{E}\cdot d\vec{A}=\frac{Q_{\mathrm{enc}}}{\varepsilon_0} \]

   \[ \oint\vec{E}\cdot d\vec{A}=\frac{Q_{\mathrm{enc}}}{\varepsilon_0} \]

2. Identify how the electric field depends on distance \(r\):
   • Inside the sphere: \(E\propto\)
   • Outside the sphere: \(E\propto\)

Summary

Changes in electric potential relate to changes in energy for charged objects. A specific electric potential value has meaning only relative to another reference point. Choosing infinity as the zero reference defines the Coulomb potential for a point charge or uniformly charged sphere:

\[ V(r)= \]

Apply

Suppose the spherical charge distribution has charge density

\[ \rho(r)=\frac{\rho_0r}{R} \]

within the object and zero for \(r>R\). Find the electric field as a function of \(r\) both inside and outside the charge distribution.

Model

Two conducting spheres of different sizes are connected by a wire.

Explore

1. Choose a color to represent the wire voltage in the model. Color the spheres using the same code.
2. Compare the two spheres by writing greater than, less than, or equal for:

   Quantity	Sphere 1 vs. Sphere 2
   Radius
   Charge

Critical Thinking & Concept Invention

1. Because the spheres are connected by a conducting wire, how should their voltages compare? Add a voltage column to the table and indicate greater than, less than, or equal.
2. Determine how charge \(Q\) and surface charge density \(\sigma\) compare for each sphere.
   • Write an expression for the voltage of each sphere using the equations derived with infinity as the reference point.
   • Use the equation to identify the charge ratio \(\frac{Q_1}{Q_2}\), then solve for \(Q_1\).
   • Using \(\sigma=Q/A\), identify the surface charge density ratio \(\frac{\sigma_1}{\sigma_2}\).

   \[ V_1=V_2= \]

   \[ \frac{Q_1}{Q_2}= \qquad Q_1= \]

   \[ \frac{\sigma_1}{\sigma_2}= \qquad \sigma_1= \]

3. How does surface charge density compare to radius of curvature for a conductor?

A greater radius of curvature results in a surface charge density __________________.