Electric Charges and Fields

Chapter overview

Electrostatics deals with electric charges at rest. Charges produce electric fields, and electric fields exert force on other charges.

1. Electric charge

Charge is a fundamental property of matter.

Types:

• Positive charge
• Negative charge

SI unit:

$$\text{coulomb }(C)$$

Quantisation of charge:

$$q=ne$$

where:

$$e=1.6\times10^{-19}\ C$$

and $n$ is an integer.

2. Conservation of charge

Total charge of an isolated system remains constant.

$$q_{\text{initial}}=q_{\text{final}}$$

3. Coulomb's law

Force between two point charges:

$$F= \frac{1}{4\pi\varepsilon_0} \frac{|q_1q_2|}{r^2}$$

Vector form:

$$\vec{F}_{12} = \frac{1}{4\pi\varepsilon_0} \frac{q_1q_2}{r^2} \hat{r}_{12}$$

where:

$$\frac{1}{4\pi\varepsilon_0}=9\times10^9\ \text{N m}^2\text{C}^{-2}$$

4. Superposition principle

Net force on a charge due to several charges is vector sum of individual forces.

$$\vec{F} = \vec{F}_1+\vec{F}_2+\vec{F}_3+\cdots+\vec{F}_n$$

5. Electric field

Electric field at a point is force experienced by a unit positive test charge.

$$\vec{E} = \frac{\vec{F}}{q_0}$$

Electric field due to point charge:

$$\vec{E} = \frac{1}{4\pi\varepsilon_0} \frac{q}{r^2} \hat{r}$$

SI unit:

$$\text{N C}^{-1} \quad\text{or}\quad \text{V m}^{-1}$$

6. Electric field lines

Properties:

• Field lines start from positive charge and end on negative charge.
• Tangent to a field line gives direction of electric field.
• Field lines never intersect.
• Closer field lines mean stronger electric field.

7. Electric dipole

An electric dipole consists of two equal and opposite charges separated by distance.

Dipole moment:

$$\vec{p}=q(2\vec{a})$$

Magnitude:

$$p=q(2a)$$

Direction is from negative charge to positive charge.

8. Electric field due to dipole

On axial line:

$$E_{\text{axial}} = \frac{1}{4\pi\varepsilon_0} \frac{2pr}{(r^2-a^2)^2}$$

For $r\gg a$:

$$E_{\text{axial}} = \frac{1}{4\pi\varepsilon_0} \frac{2p}{r^3}$$

On equatorial line:

$$E_{\text{equatorial}} = \frac{1}{4\pi\varepsilon_0} \frac{p}{(r^2+a^2)^{3/2}}$$

For $r\gg a$:

$$E_{\text{equatorial}} = \frac{1}{4\pi\varepsilon_0} \frac{p}{r^3}$$

Direction on equatorial line is opposite to $\vec{p}$.

9. Dipole in uniform electric field

Torque:

$$\vec{\tau} = \vec{p}\times\vec{E}$$

Magnitude:

$$\tau=pE\sin\theta$$

Potential energy:

$$U=-\vec{p}\cdot\vec{E}$$ $$U=-pE\cos\theta$$

10. Electric flux

Electric flux through a surface:

$$\Phi_E = \vec{E}\cdot\vec{A}$$ $$\Phi_E=EA\cos\theta$$

For non-uniform electric field:

$$\Phi_E = \int \vec{E}\cdot d\vec{A}$$

11. Gauss law

Total electric flux through a closed surface is:

$$\oint \vec{E}\cdot d\vec{A} = \frac{q_{\text{enclosed}}}{\varepsilon_0}$$

12. Applications of Gauss law

Infinite line charge

$$E= \frac{\lambda}{2\pi\varepsilon_0 r}$$

Infinite plane sheet

$$E= \frac{\sigma}{2\varepsilon_0}$$

Spherical shell

Outside shell:

$$E= \frac{1}{4\pi\varepsilon_0} \frac{q}{r^2}$$

Inside shell:

$$E=0$$

13. Solved example

Find force between charges $2\mu C$ and $3\mu C$ separated by $0.3$ m.

$$F= 9\times10^9 \frac{(2\times10^{-6})(3\times10^{-6})}{(0.3)^2}$$ $$F=0.6\ \text{N}$$

14. Common mistakes

• Forgetting vector direction in Coulomb force.
• Using $r$ instead of $r^2$ in Coulomb's law.
• Confusing electric field and electric force.
• Drawing electric field lines from negative to positive.
• Forgetting only enclosed charge matters in Gauss law.
• Using Gauss law without symmetry.

15. Quick revision