18.5 Electric Field Lines: Multiple Charges

Electric Field Lines and Forces from Multiple Charges

When multiple charges are present, each one contributes its own electric field and exerts its own force on nearby charges. The total effect at any point is found by adding up all these individual contributions. This section covers how to calculate net forces from multiple charges and how to read and sketch electric field line diagrams for common charge arrangements.

Net Force from Multiple Charges

The superposition principle is the key idea here: the net force on any charge equals the vector sum of the individual forces from every other charge in the system.

$\vec{F}_{net} = \vec{F}_1 + \vec{F}_2 + ... + \vec{F}_n$

Each individual force is calculated using Coulomb's law:

$\vec{F} = k \frac{q_1 q_2}{r^2} \hat{r}$

• $k$ is Coulomb's constant: $8.99 \times 10^9 \, \frac{N \cdot m^2}{C^2}$
• $q_1$ and $q_2$ are the charges involved
• $r$ is the distance between the two charges
• $\hat{r}$ is the unit vector pointing from one charge toward the other

How to find the net force (step by step):

1. Identify all the charges and the test charge you're analyzing.

2. For each source charge, use Coulomb's law to find the magnitude of the force on the test charge.

3. Determine the direction of each force: like charges repel, opposite charges attract.

4. Break each force into x- and y-components.

5. Add all the x-components together, then all the y-components together.

6. Combine the total x and y components to get the net force vector.

Example: Suppose a test charge of +1 nC sits 2 cm from a +3 nC charge and 3 cm from a −2 nC charge. You'd calculate the repulsive force from the +3 nC charge and the attractive force toward the −2 nC charge separately, then add them as vectors. Don't forget to convert cm to meters before plugging into Coulomb's law.

Electric Field Diagrams for Charges

Electric field lines are a visual tool that shows both the direction and relative strength of the electric field around charges.

• Field lines point away from positive charges and toward negative charges. This follows from the definition: field lines show the direction a positive test charge would be pushed.
• Where field lines are packed closely together, the field is stronger. Where they spread apart, the field is weaker.
• Field lines never cross. At every point in space, the electric field has one unique direction. If lines crossed, that point would have two directions, which isn't physically possible.

For a single positive point charge, field lines radiate outward uniformly in all directions. For a single negative point charge, field lines point inward from all directions.

For multiple charges, the total electric field at any point is the vector sum of the individual fields (superposition again). When you sketch the combined field lines:

• Lines leaving positive charges can terminate on negative charges.
• The overall pattern reflects the combined influence of all charges present.

Field Lines Between Charge Pairs

The two most common configurations are opposite-charge pairs (dipoles) and same-charge pairs. Their field line patterns look very different.

Opposite charges (dipole):

1. Field lines emerge from the positive charge and curve around to terminate on the negative charge.
2. Lines are most concentrated near each charge, where the field is strongest.
3. The pattern is symmetrical about the line connecting the two charges.
4. If the charges have equal magnitude, every field line that leaves the positive charge ends on the negative charge.

Same-sign charges (e.g., two positive charges):

1. Field lines emerge outward from both charges (or converge inward for two negative charges).
2. Lines are again most concentrated near each charge.
3. No field lines connect the two charges directly. Instead, the lines curve away from the region between them.
4. For two charges of equal magnitude and sign, the electric field is exactly zero at the midpoint between them. This happens because the two fields point in opposite directions and cancel perfectly.

Example to try: Sketch the field lines for a +4 nC and −4 nC charge separated by 3 cm. You should see lines curving from the positive charge to the negative charge, densely packed near each charge and more spread out farther away. Then sketch two positive charges (+2 nC and +6 nC) separated by 4 cm. Notice that more lines emerge from the +6 nC charge (stronger source), and there's a point between them, closer to the +2 nC charge, where the field is zero.

Electric Field Properties

Field strength at a point is the magnitude of the electric field vector there. For a single point charge, it's given by $E = k \frac{q}{r^2}$, so it drops off with the square of the distance. For multiple charges, you find the field from each charge and add them as vectors.

Equipotential surfaces are imaginary surfaces where the electric potential has the same value everywhere. They're always perpendicular to electric field lines. No work is done when a charge moves along an equipotential surface.

Gauss's law relates the total electric flux through a closed surface to the charge enclosed inside it: $\Phi_E = \frac{q_{enc}}{\epsilon_0}$. It's especially useful for calculating electric fields when the charge distribution has high symmetry (spherical, cylindrical, or planar). You'll use this more in later sections, but the core idea is that the number of field lines passing through a closed surface depends only on how much charge is inside.