Electric Fields
Field Definition
\( \vec{E}=\dfrac{\vec{F}}{q}\)
Electric field = force per unit charge produced by source charges.
E = electric field (N/C)
F = electric force on test charge (N)
q = point charge (C)
\( \vec{F}=q\vec{E} \)
Electric fields exert forces on charges placed in the field.
Positive charges accelerate along \(E\); negative charges opposite.
Point Charge Field
\(E=\dfrac{k_e q}{r^2},\qquad \vec{E}=\dfrac{k_e q}{r^2}\hat{r}\)
Electric field produced by a point charge decreases with distance squared.
\(k_e\) = Coulomb constant = \(8.99\times10^9\) N·m²/C²
q = source charge producing the field (C)
r = distance from charge to observation point (m)
\( \vec{E}_{net}=\sum \vec{E}_i \)
Total electric field at a point produced by all charges.
\( d\vec{E}=\dfrac{k_e dq}{r^2}\hat{r},\qquad \vec{E}=\int d\vec{E} \)
Used when charge is distributed continuously along an object.
dq = infinitesimal charge element.
Charge Density
\( \lambda=\dfrac{Q}{L},\qquad dq=\lambda dl \)
λ = linear charge density (C/m) — charge distributed along a line.
Q = total charge (C)
L = total length of the charged object (m)
dl = infinitesimal length element along the object
\( \sigma=\dfrac{Q}{A},\qquad dq=\sigma dA \)
σ = surface charge density (C/m²) - Charge distributed over a surface.
\( \rho=\dfrac{Q}{V},\qquad dq=\rho dV \)
ρ = volume charge density (C/m³) - Charge distributed through a volume.
Common Field Results
\( dq=\lambda dx \)
\( E=\int\dfrac{k_e\lambda}{x^2+r^2}\,dx \)
\( E=\dfrac{2k_e\lambda}{r} \)
Electric field from an infinitely long charged wire.
λ = linear charge density (C/m)
x = position along the wire
r = perpendicular distance from wire to observation point
\( dq=\lambda dl \)
\( dl=R\,d\theta \)
\( E=\int\dfrac{k_e\lambda R}{R^2+z^2}\,d\theta \)
\( E=\dfrac{k_eQz}{(z^2+R^2)^{3/2}} \)
Electric field on the axis of a charged ring.
Q = total charge on ring (C)
R = ring radius (m)
z = distance from ring center along axis (m)
θ = angular position on ring
\( dq=\sigma dA \)
\( E=\int\dfrac{k_e\sigma}{r^2}\,dA \)
\( E=\dfrac{\sigma}{2\varepsilon_0} \)
Electric field from an infinite charged plane.
σ = surface charge density (C/m²)
A = surface area (m²)
Constants and Geometry
\( k_e=\dfrac{1}{4\pi\varepsilon_0} \)
Relation between Coulomb constant and permittivity.
\( \varepsilon_0=8.85\times10^{-12}\ \text{C}^2/(N\cdot m^2) \)
Permittivity of free space — determines strength of electric fields in vacuum.
\( r=\sqrt{x^2+z^2} \)
Distance from a charge element to the observation point.
\( E_x=E\cos\theta,\qquad E_z=E\sin\theta \)
Resolve electric field vectors into components.