Charge

Electric charge

• Electric charge is quantized, that is, exists in discrete quantities which are integer multiples of the elementary charge $e=1.6022\times 10^{-19}\mathsf{C}\approx 1.6\times 10^{-19}\mathsf{C}$
  • The charge of an electron is $-e$ and the charge of a proton is $+e$
  • The SI unit of charge is the coulomb (C)
• Conservation of charge: the total charge in an isolated system remains constant
• An object can become charged by:
  • rubbing (friction)
  • conduction (transfer of charge from one charged object to another by touching)
  • induction
• $m_{\mathrm{e}}=9.11\times 10^{-31}\mathsf{kg}$ is the mass of an electron
• $n=\frac{Q}{e}$ is the number of electrons transferred

Coulomb’s Law

• $F=k\frac{q_1{q}_2}{r^2}$ is the electrostatic force (or Coulomb force) between two charges (in $\mathsf{N}$)

  • $q_1$ and $q_2$ are the magnitudes of the charges (in $\mathsf{C}$)
  • $r$ is the distance between the charges (in $\mathsf{m}$)
  • $k=\frac{1}{4\pi {ϵ}_0}=8.99\times 10^9\mathsf{N}\cdot {\mathsf{m}}^2/{\mathsf{C}}^2$ is Coulomb’s constant
  • ${ϵ}_0=\frac{1}{4\pi k}=8.85\times 10^{-12}{\mathsf{C}}^2/N\cdot {\mathsf{m}}^2$ is the permittivity of free space

• limitations and assumptions of Coulomb’s Lawtodo

  • point charges
  • objects are at rest (electrostatics force)
  • electric force

• (Superposition principle) The total force on a charge $Q$ is the sum of the forces exerted by the other charges $q_1,q_2,q_3,...$ on $Q$

  • $F=F_1+F_2+F_3+...$

• $|{\vec{F}}_E|=\frac{1}{4\pi {ϵ}_0}\frac{|q_1{q}_2|}{r^2}=k\frac{|q_1{q}_2|}{r^2}$todo

Charge Density

• $\lambda =\frac{Q}{L}$ is the linear charge density (in $C/m$)

• $\sigma =\frac{Q}{A}$ is the surface charge density (in $C/{\mathsf{m}}^2$)

• $\rho =\frac{Q}{V}$ is the volume charge density (in $C/{\mathsf{m}}^3$)

• $Q_{\text{total}}=\int \rho (r)dV$

Electric field

• An electric field is a vector field that associates to each point in space the force per unit of charge exerted on an infinitesimal test charge at rest at that point $E=\frac{F}{q}$
  • $E$ is the electric field that a charge $q$ experiences (in $N/C$)
  • $F$ is the force on a charge (in $\mathsf{N}$)
  • $q$ is the test charge (in $\mathsf{C}$)
• (vector form: $\vec{\mathbf{E}}=\frac{\vec{\mathbf{F}}}{q}$) or $\vec{\mathbf{E}}=\underset{q\to 0}{\lim }\frac{\vec{\mathbf{F}}}{q}$
• The SI unit of electric field is $N/C=V/m$

Electric Field due to a Point Charge

• $E=k\frac{Q}{r^2}⟸E=\frac{F}{q}=\frac{kQq/r^2}{q}$

  • $P$ is the point in space where the electric field is being calculated
  • $Q$ is the point charge creating the electric field (in $\mathsf{C}$)
  • $r$ is the distance between the point $P$ and the charge $Q$ (in $\mathsf{m}$)
  • $E$ is the electric field (at $P$) due to the source charge $Q$ (in $N/C$)
  • $\vec{\mathbf{E}}=k\frac{Q}{r^2}\hat{\mathbf{r}}$ where $\hat{\mathbf{r}}$ is the unit vector pointing from $Q$ to $P$
  • $k$ is Coulomb’s constant

• (Superposition Principle) The total electric field at a point in space is the vector sum of the electric fields due to the individual charges

  • ${\vec{\mathbf{E}}}_{\text{total}}={\vec{\mathbf{E}}}_1+{\vec{\mathbf{E}}}_2+{\vec{\mathbf{E}}}_3+...$

• $\vec{E}=k\int \frac{dq}{r^2}\hat{r}$todo

Notes

• There is no electric charge at point $P$. But there is an electric field there. The only real charge is $Q$.
• Notice that $E$ depends only on the charge $Q$ which produces the electric field, and not on the value of the test charge $q$.
• In the figure, the electric field is positive, so it points towards a negative charge and away from a positive charge. But if the electric field is negative, it is the opposite.

Electric Field and Potential Eifference

• $E=-\frac{V_{ba}}{x}$ is the electric field (uniform $\vec{\mathbf{E}}$)
  • $V_{ba}$ is the potential difference between points $a$ and $b$ (in $\mathsf{V}$)
  • $x$ is the distance between the points (in $\mathsf{m}$)
• $E_x=-\frac{dV}{dx}$ is the electric field (non-uniform $\vec{\mathbf{E}}$)

Electric Field between Two Parallel Plates

• $E=\frac{Q}{{\epsilon }_{0}A}$ is the magnitude of the electric field between two parallel plates, oppositely charged
  • $Q$ is the charge on each plate
  • $A$ is the area of one plate (Gaussian surface)
  • Given the plate separation is much smaller than the dimensions of the plates
  • This equation is derived from Gauss’s Law and the principle of superposition:

Electric Field Lines

• Electric field lines indicate the direction of the electric field; the field points in the direction tangent to the field line at any point (note that the field lines never cross)

• The lines are drawn such that the magnitude of the electric field, $E$, is proportional to the number of lines crossing unit area perpendicular to the lines. The closer the lines, the stronger the field

• The lines start on positive charges and end on negative charges

Electric dipole moment

• An electric dipole consists of two equal and opposite charges separated by a distance $d$
• The electric dipole moment $\vec{\mathbf{p}}$ (in $\mathsf{C}\cdot \mathsf{m}$) of a dipole is defined as $\vec{\mathbf{p}}=q\vec{\mathbf{d}}$ where $q$ is the magnitude of the charges and $\vec{\mathbf{d}}$ is the vector pointing from the negative charge to the positive charge
• $\vec{\mathbf{\tau }}=\vec{\mathbf{p}}\times \vec{\mathbf{E}}$ is the torque on an electric dipole
• $U=-\vec{\mathbf{p}}\cdot \vec{\mathbf{E}}$ is the potential energy for an electric dipole in an electric field

Electric flux

• ${\Phi }_E=\oint \vec{\mathbf{E}}\cdot d\vec{\mathbf{A}}$ is the electric flux through a closed surface
  • $\oint$ indicates a closed surface integral
  • $d\vec{A}$ is the differential area vector pointing outward from the surface (in ${\mathsf{m}}^2$)
  • $\vec{E}\cdot d\vec{A}$ is the dot product of the electric field and the area vector
  • (uniform electric field) ${\Phi }_E=EA\cos \theta$ where $A$ is the area of the surface and $\theta$ is the angle between the electric field and the normal to the surface
• (Gauss’s Law)
  • (integral form) ${\Phi }_E=\oint \vec{\mathbf{E}}\cdot d\vec{\mathbf{A}}=\frac{Q_{\text{enc}}}{{\epsilon }_0}$ is the electric flux through a closed surface where $Q_{\text{enc}}$ is the total charge enclosed by the surface
  • (differential form) $\nabla \cdot \mathbf{E}=\frac{\rho }{{\epsilon }_0}$, where $\rho$ is the charge density (in $C/{\mathsf{m}}^3$)