Electric field of a point charge
How strong, how far?
- Close to a single charge the field is intense; step back and it weakens fast.
- A clean formula tells us exactly how fast.
- And it mirrors gravity almost perfectly.
Field of a point charge
- At distance $r$ from a point charge $Q$: $E = \dfrac{Q}{4\pi\varepsilon_0 r^{2}}$.
- It points out from a positive charge, in toward a negative one, and falls as $\dfrac{1}{r^{2}}$.
The field strength at distance $r$ from a point charge $Q$ is:
From $E = F/q$ with Coulomb's force, the test charge cancels, leaving $E = \dfrac{Q}{4\pi\varepsilon_0 r^{2}}$.
The field is $100\ \dfrac{\text{N}}{\text{C}}$ at distance $r$ from a charge. What is it at $2r$?
Inverse-square: $\dfrac{100}{2^{2}} = 25\ \dfrac{\text{N}}{\text{C}}$.
The field points ____ from a positive point charge.
A positive charge pushes a positive test charge away, so its field points outward.
Just like gravity
- Compare $E = \dfrac{Q}{4\pi\varepsilon_0 r^{2}}$ with $g = \dfrac{GM}{r^{2}}$ — the same inverse-square shape.
- The difference: charge can be ±, so the field can attract or repel; gravity only attracts.
The electric field of a point charge falls off with distance just like gravity (as 1/r²).
Both are inverse-square: $E = \dfrac{Q}{4\pi\varepsilon_0 r^{2}}$ and $g = \dfrac{GM}{r^{2}}$.
Many charges
- For several charges, the total field is the vector sum of each one's field.
- Add them as arrows (size and direction), not just numbers.
For several charges, the total field at a point is:
Fields are vectors, so add them with both size and direction.
You've got it
- field of a point charge: $E = \dfrac{Q}{4\pi\varepsilon_0 r^{2}}$ (out from +, in to −)
- it falls as $\dfrac{1}{r^{2}}$ — the same shape as gravity
- combine several fields by vector sum