The Electric Field

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 🌐 Version en français : Champ électrique

📘 The Electric Field

Reference Textbook

📖 Reference Textbook:
University Physics Volume 2: Chapter 5.4

Theory

The Electric Field

The electric field provides a convenient way of calculating the electric force $\overrightarrow{F}$ on a test charge $q_{\mathrm{t}}$. It plays a role similar to that of the gravitational field $\overrightarrow{g}$ when calculating gravitational forces.

Instead of recalculating the electric force directly using Coulomb's law—and adding the individual forces exerted by several source charges—we can first determine the electric field:

$\overrightarrow{F}=q_{\mathrm{t}}\overrightarrow{E}$

where:

• $\overrightarrow{F}$ is the electric force exerted on the test charge;
• $q_{\mathrm{t}}$ is the test charge placed in the electric field;
• $\overrightarrow{E}$ is the electric field at the position of the test charge.

In the same way that the gravitational field $\overrightarrow{g}$ gives the gravitational force per unit mass, the electric field $\overrightarrow{E}$ gives the electric force per unit charge:

$\overrightarrow{E}=\frac{\overrightarrow{F}}{q_{\mathrm{t}}}$

The SI unit of electric field is the newton per coulomb:

$[\overrightarrow{E}]=\mathrm{N}/\mathrm{C}$

▶ Video: Introduction to the Electric Field

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Electric Field of a Point Charge

The electric field $\overrightarrow{E}$ created by a stationary point charge $q_{\mathrm{s}}$ at a distance $r$ is given by:

$\overrightarrow{E}=k\frac{q_{\mathrm{s}}}{r^2}\hat{r}$

where:

• $\overrightarrow{E}$ is the electric field created by the source charge;
• $k$ is Coulomb's constant, approximately $8.99\times 10^9\mathrm{N}\mathrm{\cdot }{\mathrm{m}}^{\mathrm{2}}/{\mathrm{C}}^{\mathrm{2}}$;
• $q_{\mathrm{s}}$ is the source charge creating the electric field;
• $r$ is the distance from the source charge to the point where the field is calculated;
• $\hat{r}$ is a unit vector pointing from the source charge toward the point of interest.

The direction of the electric field depends on the sign of the source charge:

• The electric field points away from a positive source charge.
• The electric field points toward a negative source charge.

Combining the electric-field equation with

$\overrightarrow{F}=q_{\mathrm{t}}\overrightarrow{E}$

gives Coulomb's law:

${\overrightarrow{F}}_{\mathrm{t}\mathrm{s}}=k\frac{q_{\mathrm{s}}{q}_{\mathrm{t}}}{r^2}\hat{r}$

▶ Video: Electric Field of a Point Charge

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Superposition of Electric Fields

When several source charges are present, the total electric field at a point is the vector sum of the electric fields created by the individual charges:

${\overrightarrow{E}}_{\mathrm{n}\mathrm{e}\mathrm{t}}=\sum _i{\overrightarrow{E}}_i$

For a collection of point charges:

${\overrightarrow{E}}_{\mathrm{n}\mathrm{e}\mathrm{t}}=\sum _{i}k\frac{q_i}{r_i^2}{\hat{r}}_i$

The electric field is calculated independently for each source charge. The resulting electric-field vectors are then added using vector addition.

Electric Field Lines

Electric field lines provide a visual representation of an electric field. At any point, the electric-field vector is tangent to the field line and points in the direction of the force that would be exerted on a positive test charge.

• Direction: Field lines point away from positive charges and toward negative charges.
• Field strength: The density of field lines represents the magnitude of the electric field. Closely spaced lines indicate a stronger field.
• Positive test charge convention: The arrows show the direction in which a positive test charge would accelerate.

Rules for Drawing Electric Field Lines

• Field lines begin on positive charges and end on negative charges or at infinity.
• The number of field lines associated with a charge is proportional to the magnitude of the charge.
• Electric field lines never cross.
• Field lines form continuous, smooth curves.
• Field lines do not form closed loops in electrostatics.
• Near the surface of a conductor in electrostatic equilibrium, field lines are perpendicular to the surface.

▶ Video: Electric Field Lines

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Typical Electric-Field Problems

Typical electric-field problems may require you to:

• calculate the electric field created by one or more point charges;
• determine the direction of the electric field at a specified point;
• add electric-field vectors using the principle of superposition;
• determine the force exerted on a test charge placed in an electric field;
• find a position where the net electric field is zero.

A useful general approach is:

1. Draw a diagram showing all source charges and the point of interest.
2. Determine the electric field created by each source charge.
3. Establish the direction of each electric-field vector.
4. Resolve the vectors into components when necessary.
5. Add the components to obtain the net electric field.
6. Use $\overrightarrow{F}=q_{\mathrm{t}}\overrightarrow{E}$ if the force on a test charge is required.

▶ Video: Typical Problems with the Electric Field

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Demonstrations and Additional Videos

Faraday Cage

A Faraday cage is a conducting enclosure that redistributes electric charges around its exterior surface. Under electrostatic conditions, the electric field inside the conducting material and within an enclosed empty cavity is zero, provided that no charge is located inside the cavity.

▶ Demonstration: MIT Physics Demo — Faraday's Cage

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Additional Electric-Field Videos

▶ Additional Video 1

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▶ Additional Video 2

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Electric-Field Simulations

Explore electric fields with these simulations:

• PhET: Electric Field of Dreams
• PhET: Charges and Fields
• PhET: Electric Field Hockey

Additional simulations involving time-varying electric fields and electromagnetic waves:

• PhET: Radio Waves and Electromagnetic Fields
• PhET: Microwaves

Additional Resources

• Electric Field Vectors
• Three-Dimensional Electric-Field Visualization
• Lightning Applet

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