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 $\mathrm{N}$)
  • $q$ is the test charge (in $\mathrm{C}$)
• (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

• $E=k\frac{Q}{r^2}$

  • $P$ is the point in space where the electric field is being calculated
  • $Q$ is the point charge creating the electric field (in $\mathrm{C}$)
  • $r$ is the distance between the point $P$ and the charge $Q$ (in $\mathrm{m}$)
  • $E$ 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 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

Electrostatic field

• An electrostatics field (or static electric field) is an electric field that does not change with time
• For any electrostatic field:
  • $\vec{\mathbf{E}}=-\nabla V$
    • (uniform electrostatic field) $E=-\frac{V_{ba}}{d}$
      • $V_{ba}$ is the potential difference between points $a$ and $b$ (in $\mathrm{V}$)
      • $d$ is the distance between the points (in $\mathrm{m}$)
  • ${\oint }_{\mathcal{C}}\vec{\mathbf{E}}\cdot d\vec{\mathbf{\ell }}=0$ for any closed curve $\mathcal{C}$

Electric flux

• ${\Phi }_E={\iint }_S\vec{\mathbf{E}}\cdot d\vec{\mathbf{S}}$ is the electric flux (in $\mathrm{N}\cdot {\mathrm{m}}^2/C$) through a surface $S$ in an electric field $\vec{\mathbf{E}}$
  • ${\Phi }_E=\vec{\mathbf{E}}\cdot \vec{\mathbf{A}}$ is the electric flux through a flat surface with a vector area $\vec{\mathbf{A}}$ in a uniform electric field $\vec{\mathbf{E}}$.
  • (See Gauss’s Law for electricity)

Electric displacement field

• electric displacement field (or electric flux density)
• $\vec{\mathbf{D}}={\epsilon }_0\vec{\mathbf{E}}+\vec{\mathbf{P}}$
  • $\vec{\mathbf{P}}$ is the polarization density
• displacement current density $\frac{\partial \vec{\mathbf{D}}}{\partial t}$