5.5: Conductors in Electric Fields

Learning Objectives

• Predict how an electric field will influence the motion of positive and negative charges.
• Construct qualitative models to show the equilibrium state for charges in/on a conductor in the presence of a zero or nonzero external electric field.

Model

Explore

1. When an electric field is applied to a conductor, what moves? What direction do they move?
2. Describe the charge distribution (i.e., where they are located) when an external electric field is applied to a conductor.
3. What is the electric field inside the middle of the conductor?

Critical Thinking & Concept Invention

1. What direction would the electric field from the positive and negative charges point? Hint: It is not zero.
2. How is it possible that the electric field in a conductor is zero, given your answer to the previous question?

Model

The box below is a conductor and the arrows represent electric field vectors at the locations shown. Recall that charges can move freely inside a conductor but cannot leave a conductor.

Explore

1. Next to each electric field vector, draw an arrow to identify the direction of force that the charge experiences. Hint: Pay attention to the sign of the charge.
2. Forces will cause stationary charges to accelerate and therefore move. Consider what direction the charges will be pushed and draw where each charge might be a few moments later.
3. Circle any charges that were not able to move in the direction of the force on that/those charge(s).

Critical Thinking & Concept Invention

1. For the charges that do not move: Is the force parallel or perpendicular to the surface of the conductor? (One might be tricky to determine.)
2. For the charges on the surface that do move, is it the parallel or perpendicular component of the electric field, with respect to the surface, that points in the direction (or opposite) that the charge moves?
3. Summarize the patterns by choosing the correct selection:
   • A component of the electric field that is parallel to the surface of a conductor WILL / WILL NOT cause a charge to move.
   • A component of the electric field that is perpendicular to the surface of a conductor WILL / WILL NOT cause a charge to move.
4. Unless an external electric field is changing, charges will move until equilibrium, then remain still. At equilibrium, which of the following must be true:

\[ E_{\parallel}=0 \quad \text{OR} \quad E_{\perp}=0 \]

5. If charges in the interior of the conductor also do not move, what must be true about the electric field inside a conductor?

\[ E_{\text{internal}}= \]

Model

A conductor with net charges shown does not interact with any external object or fields. The arrows shown represent the electric fields created by charges in/on that conductor only. Six different closed surfaces are shown (imagine the ones that appear two-dimensional are in fact closed three-dimensional objects of those arbitrary shapes).

Explore

1. Is the electric field zero or nonzero inside the conductor?
2. Identify which areas (if any) have:
   • Positive enclosed charge
   • Negative enclosed charge
   • Neutral or no enclosed charge
   • Positive electric flux
   • Negative electric flux
   • No electric flux

Critical Thinking & Concept Invention

Examine Areas 2, 3, and 4:

1. Is it possible that there could be a positive charge in any of those areas? Explain why or why not.
2. Is it possible that there could be a negative charge in any of those areas? Explain why or why not.
3. Construct a rule about net charge inside a conductor. Explain that rule.

Model

Both objects are conductors. The cavity is an empty, closed space within the conductor and the charges inside those cavities do not touch the conductor itself.

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

Explore

1. Draw electric field lines from each charge inside the cavity. Remember that each line should originate on a positive charge and terminate on a negative charge.
2. Draw electric field lines from each charge (if present) on the surface of the conductor.
3. Identify, based on the number of lines, the flux inside each cavity for a surface that encloses the charge but not the charges along the surface of the cavity.
4. Identify, based on the number of lines, the flux outside of the full conductor in each example.

Critical Thinking & Concept Invention

1. How do the cavity fluxes and whole-conductor fluxes compare?
2. In the cavity, is the surface charge distributed evenly or unevenly? If uneven, where is there more concentrated charge?
3. On the surface of the conductor as a whole, is the surface charge distributed evenly or unevenly? If uneven, where is there more concentrated charge?
4. If we cannot see the cavity charges, can we tell how much net charge there is inside the conductor based on the surface charge of the conductor? Explain.
5. If we cannot see the cavity charges, can we tell how or where the charge inside the conductor is distributed spatially based on the surface charge of the conductor? Explain.

Apply

Three charges are present within the cavity of a conductor.

• Draw electric field lines inside the cavity, inside the conductor, and outside of the conductor. If the electric field is zero in any of these regions, write: \[ \vec{E}=0 \]
• Draw charges to show how they will be distributed on both surfaces of the conductor (cavity and outer surface) and inside the conductor. If there is zero net charge in/on any of these regions, write: \[ Q=0 \]

Model

A neutral conductor is shown on the left. Positive charge is added and the right image shows the distribution of charge at equilibrium. The small positive signs on the right represent a fraction of the charge added (large positive sign in the left image). The dashed lines represent theoretical closed surfaces for the purpose of electric flux models.

Explore

1. How much charge (variable, not number) is added to the conductor?
2. How much charge ends up within the inner dashed surface?
3. How much charge ends up within the outer dashed surface?
4. Is the charge distributed evenly or unevenly in/on the conductor? If uneven, describe how the charge is distributed.

Critical Thinking & Concept Invention

1. Draw an electric field line from each positive charge. Recall the rules established for how those lines should be drawn (e.g., inside/outside the conductor and angle to the surface).
2. How much of the total charge is:
   • Inside the conductor?
   • On the surface of the conductor?
3. Construct a rule:
   • Surface charge density is greater (i.e., charges are closer together) __________________.
4. Identify the electric flux (in terms of given variables) for both dashed surfaces.

Apply

The two shapes below are conductors. Both conductors have a total charge of \(-Q\).

• Draw charges (+ or − signs) to show the general distribution of charges.
• Draw electric field lines or write \[ \vec{E}=0 \] to represent the electric field inside and outside of the conductors.

Inside the Conductor

• The net charge is: __________________
• Because of this, the electric field is: __________________

On the Surface of the Conductor

• If the conductor is neutral, the net surface charge is: __________________
• If the conductor is charged, the net surface charge is: __________________
• The distribution of charge is even if: __________________
• The density of charge is greater at locations: __________________
• The orientation of the electric field is: __________________