AP Physics C E&M- 8.4 Electric Fields Charge Distributions- Study Notes- New Syllabus

Electric Field Resulting from a Given Charge Distribution

The electric field at a point in space due to a charge distribution is the vector sum of contributions from all individual charges that make up the distribution. Depending on whether the charge is localized or spread out, the distribution can be discrete (point charges) or continuous (line, surface, or volume charges).

Superposition Principle (Discrete Charges):

For a system of discrete point charges:

\( \mathrm{\vec{E}_{net} = \sum_i \vec{E}_i = \sum_i \dfrac{k_e q_i}{r_i^2} \hat{r}_i} \)

Continuous Charge Distributions (Integral Form):

• When charges are continuously distributed, the electric field is found by integration:

1. Line Charge:

For linear charge density \(\mathrm{\lambda = \dfrac{dq}{dl}}\):

\( \mathrm{\vec{E} = \dfrac{1}{4 \pi \varepsilon_0} \int \dfrac{\lambda \, dl}{r^2} \, \hat{r}} \)

2. Surface Charge:

For surface charge density \(\mathrm{\sigma = \dfrac{dq}{dA}}\):

\( \mathrm{\vec{E} = \dfrac{1}{4 \pi \varepsilon_0} \int \dfrac{\sigma \, dA}{r^2} \, \hat{r}} \)

3. Volume Charge:

For volume charge density \(\mathrm{\rho = \dfrac{dq}{dV}}\):

\( \mathrm{\vec{E} = \dfrac{1}{4 \pi \varepsilon_0} \int \dfrac{\rho \, dV}{r^2} \, \hat{r}} \)

Symmetry and Gauss’s Law(Topic – 8.6):

For highly symmetric charge distributions (spherical, cylindrical, planar), Gauss’s law provides a more direct method:

\( \mathrm{\oint \vec{E} \cdot d\vec{A} = \dfrac{Q_{enclosed}}{\varepsilon_0}} \)

Key Idea: For complex or asymmetric charge distributions, integration is required. For symmetric distributions, Gauss’s law greatly simplifies the calculation.