18.3 Electric Field

Electric Field

Electric fields describe how a charge influences the space around it. Any other charge placed in that space will feel a force. The electric field gives you a way to predict the strength and direction of that force at every point, even before you place a second charge there.

This section covers how to calculate electric field strength and direction, how to read and draw field line diagrams, and how the superposition principle lets you handle situations with multiple charges.

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Electric Field

Strength and Direction of Electric Fields

The electric field ($\vec{E}$) is a vector quantity that describes the force a positive test charge would experience per unit charge at any point in space. It's measured in N/C (or equivalently, V/m).

For a point charge $q$, the electric field at a distance $r$ is:

$\vec{E} = \frac{kq}{r^2}\hat{r}$

• $k$ is Coulomb's constant: $k = 8.99 \times 10^9 \text{ N} \cdot \text{m}^2 / \text{C}^2$
• $\hat{r}$ is the unit vector pointing from the source charge to the point where you're measuring the field

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

• Positive charges create fields that point radially outward (away from the charge)
• Negative charges create fields that point radially inward (toward the charge)

A useful way to remember this: the field always points in the direction a positive test charge would be pushed.

Field strength follows an inverse-square relationship ($1/r^2$). If you double your distance from a charge, the field strength drops by a factor of 4. Triple the distance, and it drops by a factor of 9.

Two more things worth knowing:

• Outside a uniformly charged sphere, the electric field behaves exactly as if all the charge were concentrated at the center. This is why you can treat charged spheres as point charges for external calculations.
• The force on a charge $q_0$ placed in an external field is $\vec{F} = q_0 \vec{E}$. This is the direct link between electric field and electrostatic force.

Electric Field Line Diagrams

Field line diagrams give you a visual map of the electric field in a region. They don't show exact numbers, but they reveal the field's direction and relative strength at a glance.

Rules for reading and drawing field lines:

• Field lines originate on positive charges and terminate on negative charges
• The density of lines (how closely packed they are) indicates field strength. Tightly packed lines mean a stronger field.
• Field lines never cross. If they did, it would imply two different field directions at the same point, which is physically impossible.

Single point charge: Lines extend radially outward from a positive charge and radially inward toward a negative charge. The lines spread apart as you move farther away, reflecting the weakening field.

Electric dipole (two equal and opposite charges): Lines curve from the positive charge to the negative charge. They're most concentrated near the charges themselves, where the field is strongest. A water molecule ($H_2O$) is a common real-world example of a dipole.

Two equal charges of the same sign: Lines extend outward from both charges. In the region between them, the lines push apart and the field is weaker. At the exact midpoint between two identical charges, the fields cancel completely and $\vec{E} = 0$.

Field line diagrams are qualitative tools. They show patterns and relative strengths, but you need the equation to get actual magnitudes.

Electric flux is a related concept: it measures how much electric field passes through a given surface area. You can think of it as "counting" how many field lines pierce a surface.

Superposition of Multiple Charges

The principle of superposition states that the net electric field at any point equals the vector sum of the individual fields produced by each charge. Charges create their fields independently, and those fields simply add together.

To find the net electric field at a point due to multiple charges:

1. Identify all source charges and the point where you want to find the field.
2. Calculate the electric field at that point due to each charge individually using $\vec{E} = \frac{kq}{r^2}\hat{r}$.
3. Determine the direction of each individual field (away from positive charges, toward negative charges).
4. Add the individual field vectors using vector addition: $\vec{E}_{net} = \vec{E}_1 + \vec{E}_2 + ...$

Since these are vectors, you can't just add magnitudes. Break each field into x- and y-components, sum the components separately, then recombine to find the magnitude and direction of the resultant. This is where your trigonometry skills become essential.

Superposition works for any number of charges and any arrangement. You can apply it to simple two-charge problems, or extend it to more complex distributions like lines or planes of charge. It's the same principle at work in practical devices like capacitors and particle accelerators.

Work and Potential Energy in Electric Fields

When a charge moves through an electric field, the field can do work on it, and that work changes the charge's electric potential energy.

• Electric potential energy is the energy stored in a system of charges due to their relative positions. Two like charges pushed close together have high potential energy (they "want" to fly apart). Two opposite charges pulled apart also have high potential energy (they "want" to come together).
• Equipotential surfaces are imaginary surfaces where the electric potential has the same value everywhere. Field lines are always perpendicular to equipotential surfaces.
• Moving a charge along an equipotential surface requires no work, because there's no component of the electric field in that direction. Moving a charge across equipotential surfaces (parallel to field lines) does require work against or by the field.