Electrostatics - Charges, Forces, Fields, and Gauss’s Law
- Page ID
- 147289
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Learning Objectives
- Describe electric charge, its properties, and conservation
- Distinguish conductors vs insulators and charging methods
- Apply Coulomb’s Law and superposition
- Define and compute electric fields
- Calculate fields for discrete and continuous charge distributions
- Use electric flux and Gauss’s Law
- Apply Gauss’s Law to symmetric systems
- Describe conductors in electrostatic equilibrium
1. Electric Charge
Key Ideas
- Two types: positive (+) and negative (−)
- Like charges repel; opposite charges attract
- Charge is quantized:
\( q = ne,\quad e = 1.6\times10^{-19} \, C \) - Charge is conserved
Atoms are neutral; charging occurs by transfer of electrons.
Worked Examples
For conducting spheres to get a charge of +4 μC, electrons have to be removed. How many electrons need to be removed from the sphere?
Solution
Total charge q = 4 μC.
Since the charge is quantized, we can write the total charge as:
\( q = N e\) where \( e= 1.6\times10^{-19} \, C \) is the quantum of charge and \( N \) is the number of electrons removed.
Two identical conducting spheres touch; one has +4 μC, the other is neutral. What is the final charge in each of the spheres?
Solution
Total charge = 4 μC. After contact, charge splits equally:
\( q = 2\,\mu C \) on each sphere.
Additional Details (Optional)
Electrons are mobile and responsible for most electrostatic effects; protons remain bound in nuclei.
2. Conductors, Insulators, and Charging
- Conductors: free electrons move easily
- Insulators: electrons are bound
- Charging methods:
- Conduction (contact)
- Induction (no contact)
- Friction (triboelectric effect)
Worked Example
Charging by induction
Click to reveal solution
- Bring charged rod near conductor
- Charges redistribute
- Ground conductor → charge flows
- Remove ground, then rod
Result: object acquires opposite charge
Additional Details (Optional)
Triboelectric effects depend on material properties, surface conditions, and environment.
3. Coulomb’s Law & Superposition
\[ \vec{F} = k \frac{q_1 q_2}{r^2} \hat{r}, \quad k = 9\times10^9 \]
- Force is vector-valued
- Superposition principle: \[ \vec{F}_{net} = \sum \vec{F}_i \]
Worked Examples
Force between electron and proton
Click to reveal solution
\[ F = k \frac{(1.6\times10^{-19})^2}{(10^{-10})^2} = 2.3\times10^{-8} N \]
Multiple charges (vector addition)
Click to reveal solution
Compute each force, resolve into components:
\( F_x = \sum F_i \cos\theta, \quad F_y = \sum F_i \sin\theta \)
Combine components to obtain net force vector.
4. Electric Field
\[ \vec{E} = \frac{\vec{F}}{q}, \quad \vec{E} = k \frac{Q}{r^2} \hat{r} \]
- Units: N/C
- Direction defined by positive test charge
Worked Examples
Field from a point charge
Click to reveal solution
\( E = kQ/r^2 \)
Force from a field
Click to reveal solution
\( F = qE \). If \( q<0 \), force is opposite to field.
Additional Details (Optional)
Electric field plays a role analogous to gravitational field \( g \), but depends on charge instead of mass.
5. Fields of Charge Distributions
Discrete: \[ \vec{E} = \sum \vec{E}_i \] Continuous: \[ \vec{E} = \int k \frac{dq}{r^2} \]
- Linear: \( \lambda = Q/L \)
- Surface: \( \sigma = Q/A \)
- Volume: \( \rho = Q/V \)
Strategy
- Divide into \( dq \)
- Express using density
- Integrate
- Exploit symmetry
Worked Examples
Line of charge
Click to reveal solution
Use \( dq=\lambda dx \), integrate along length, symmetry cancels horizontal components.
Infinite plane
Click to reveal solution
\( E = \frac{\sigma}{2\epsilon_0} \)
6. Electric Flux
\[ \Phi_E = \vec{E}\cdot\vec{A} = EA\cos\theta \]
Worked Example
Click to reveal solution
Maximum flux when field is perpendicular; zero when parallel.
7. Gauss’s Law
\[ \oint \vec{E}\cdot d\vec{A} = \frac{Q_{enc}}{\epsilon_0} \]
Using Gauss’s Law
- Choose symmetric Gaussian surface
- Ensure \( E \) is constant or zero over parts of surface
Worked Examples
Point charge
Click to reveal solution
\( E = kq/r^2 \)
Infinite line
Click to reveal solution
\( E = \frac{\lambda}{2\pi\epsilon_0 r} \)
Infinite plane
Click to reveal solution
\( E = \frac{\sigma}{2\epsilon_0} \)
8. Conductors in Electrostatic Equilibrium
- Electric field inside conductor = 0
- Charge resides on surface
- Field just outside: \[ E = \frac{\sigma}{\epsilon_0} \]
- Field is perpendicular to surface
Worked Example
Charged conducting sphere
Click to reveal solution
Outside: \( E = kQ/r^2 \)
Inside: \( E = 0 \)
Additional Details (Optional)
Charge accumulates more strongly at sharp points, leading to stronger electric fields.
Summary of Key Equations
- \( F = k \frac{q_1 q_2}{r^2} \)
- \( E = k \frac{Q}{r^2} \)
- \( F = qE \)
- \( \Phi = EA\cos\theta \)
- \( \Phi = \frac{Q_{enc}}{\epsilon_0} \)
Electrostatics: Charges, Forces, and Fields
1. Origin and Properties of Charge
Charge is an intrinsic property of matter. There are two types:
- Positive charge (protons)
- Negative charge (electrons)
Key properties:
- Like charges repel
- Opposite charges attract
- Atoms are electrically neutral
Typical scales:
- Nucleus radius: \(10^{-15}\,m\)
- Atom radius: \(10^{-10}\,m\)
---
2. Charge Quantization
Charge comes in discrete units:
\[ q = n e \]
where:
- \( e = 1.6 \times 10^{-19} \, C \)
- \( n \) is an integer
---
3. Coulomb’s Law
The force between two charges is given by:
\[ \vec{F} = k \frac{q_1 q_2}{r^2} \hat{r} \]
where:
- \( k = 9\times10^9 \, \mathrm{N\cdot m^2/C^2} \)
---
Example: Electron-Proton Force
Show solution
\[ F = k \frac{(1.6\times10^{-19})^2}{(10^{-10})^2} = 2.3\times10^{-8} \, N \]---
4. Superposition of Forces
The net force is the vector sum:
\[ \vec{F}_{net} = \sum \vec{F}_i \] 
---
Example: Three Charges
Show solution
- Compute each force using Coulomb’s law
- Resolve into x and y components
- Sum components
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5. Conductors and Insulators
- Conductors: electrons move freely
- Insulators: electrons are bound
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6. Charging Methods
Conduction
Charge transferred by contact
Induction
Charge rearranged without contact
---
7. Electric Field
Electric field is defined as:
\[ \vec{E} = \frac{\vec{F}}{q} \]
For a point charge:
\[ \vec{E} = k \frac{Q}{r^2} \hat{r} \] 
---
Relation Between Force and Field
\[ \vec{F} = q\vec{E} \] ---
Example
Show solution
Substitute into \( E = kQ/r^2 \), then compute force using \( F = qE \).
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8. Electric Dipole
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9. Continuous Charge Distributions
Charge densities:
- Linear: \( \lambda = \frac{Q}{L} \)
- Surface: \( \sigma = \frac{Q}{A} \)
- Volume: \( \rho = \frac{Q}{V} \)
Electric field:
\[ \vec{E} = \int k \frac{dq}{r^2} \] ---
Procedure
- Break into \( dq \)
- Express using density
- Integrate
- Use symmetry
---
10. Electric Flux
\[ \Phi_E = \vec{E}\cdot\vec{A} = EA\cos\theta \] 
---
11. Gauss’s Law
\[ \oint \vec{E}\cdot d\vec{A} = \frac{Q_{enc}}{\epsilon_0} \] 
---
Symmetry Cases
- Sphere → spherical symmetry
- Line → cylindrical symmetry
- Plane → planar symmetry
---
Example: Infinite Plane
Show solution
\[ E = \frac{\sigma}{2\epsilon_0} \]---
12. Conductors in Electrostatic Equilibrium
- \(E = 0\) inside conductor
- Charge resides on surface
- Field perpendicular to surface
---
Example
Show solution
Inside: \( E = 0 \)
Outside: \( E = kQ/r^2 \)
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13. Motion of Charged Particles
\[ F = qE = ma \] \[ a = \frac{qE}{m} \] 
---
14. Summary of Key Equations
- \( F = k \frac{q_1 q_2}{r^2} \)
- \( E = k \frac{Q}{r^2} \)
- \( F = qE \)
- \( \Phi = EA\cos\theta \)
- \( \oint \vec{E}\cdot d\vec{A} = \frac{Q}{\epsilon_0} \)