Section 2.3: Field of Multiple Charges

This section applies the principle of superposition to calculate the net electric field produced by multiple point charges at a given location.

Superposition Principle:

\( \vec{E}_{net} = \sum_i \vec{E}_i \)

The net field is the vector sum of individual fields from each charge.

Example 1

Two charges, \( q_1 = +2\mu C \) at \( x = 0 \), and \( q_2 = -2\mu C \) at \( x = 1 \, \text{m} \). Find the net field at the midpoint.

Distance from midpoint to each charge: \( r = 0.5 \, \text{m} \).

\( E = k \frac{q}{r^2} = 8.99 \times 10^9 \frac{2 \times 10^{-6}}{(0.5)^2} \approx 7.2 \times 10^4 \, \text{N/C} \).

Both fields point in the same direction (toward the negative charge). Net field: \( 1.44 \times 10^5 \, \text{N/C} \) toward \( q_2 \).

Two equal charges \( +1\mu C \) are placed 2 m apart. Find the net field at a point midway between them.
Three charges are placed at the vertices of an equilateral triangle. Find the net field at the centroid.
Explain why the field due to two equal charges of opposite sign cancels at infinity.
A line has two charges: \( +4\mu C \) at \( x=0 \) and \( +2\mu C \) at \( x=1 \). Find the net field at \( x=0.5 \).
Conceptual: Why must electric fields be added vectorially instead of algebraically?