AcademyElectric Charge and Fields
Academy
Calculating Electric Fields
Level 1 - Physics topic page in Electric Charge and Fields.
Principle
Electric fields are calculated by adding the vector field contributions from each source charge.
Notation
\(Q_i\)
source point charge
\(\mathrm{C}\)
\(\vec r\)
field point position
\(\mathrm{m}\)
\(\vec r_i\)
source charge position
\(\mathrm{m}\)
\(\vec R_i\)
vector from source \(i\) to field point
\(\mathrm{m}\)
\(\hat R_i\)
unit vector from source \(i\) to field point
1
\(dq\)
small source charge element
\(\mathrm{C}\)
\(\lambda\)
linear charge density
\(\mathrm{C\,m^{-1}}\)
Method
Derivation 1: Field from one point charge
Start from Coulomb's force on a positive test charge and divide by the test charge.
Source-to-field vector
\[\vec R=\vec r-\vec r_Q\]
Force on test charge
\[\vec F=k\frac{Qq_0}{R^2}\hat R\]
Divide by test charge
\[\vec E=\frac{\vec F}{q_0}=k\frac{Q}{R^2}\hat R\]
Derivation 2: Add point-charge fields
The electric field is a vector. Each source creates a field at the same point, and the net field is the vector sum.
Single source contribution
\[\vec E_i=k\frac{Q_i}{R_i^2}\hat R_i\]
Superposition
\[\vec E_{\mathrm{net}}=\sum_i\vec E_i\]
Components
\[\vec E_{\mathrm{net}}=(\sum E_x)\hat\imath+(\sum E_y)\hat\jmath\]
Derivation 3: Replace a sum by an integral
For a continuous charge distribution, split the source into small charge elements and sum their fields by integration.
Small contribution
\[d\vec E=k\frac{dq}{R^2}\hat R\]
Line element
\[dq=\lambda\,dl\]
Continuous field
\[\vec E=\int k\frac{dq}{R^2}\hat R\]
Rules
These are the compact field-calculation tools.
Point-charge field
\[\vec E=k\frac{Q}{R^2}\hat R\]
Field superposition
\[\vec E_{\mathrm{net}}=\sum_i \vec E_i\]
Continuous source
\[d\vec E=k\frac{dq}{R^2}\hat R\]
Line charge
\[dq=\lambda\,dl\]
Examples
Question
Find the field magnitude
\[0.30\,\mathrm{m}\]
from a \[+6.0\,\mathrm{nC}\]
point charge.Answer
\[E=k\frac{|Q|}{r^2}=\frac{(8.99\times10^9)(6.0\times10^{-9})}{0.30^2}=6.0\times10^2\,\mathrm{N\,C^{-1}}\]
The field points away from the positive charge.Checks
- The unit vector points from each source charge to the field point.
- A negative source charge reverses the direction of its field contribution.
- Add fields as vectors, not as unsigned magnitudes.
- Symmetry can cancel components before doing algebra.