Electricity Charge and Field

Electric Charge¶

There are three kinds of charge density: - Linear charge density: \(\lambda = dq/dx\) - Surface charge density: \(\sigma = dq/dA\) - Volume charge density: \(\rho = dq/dV\)

The force between a point charge and a ring of charge is \(\(F_z = \frac{1}{4\pi \epsilon_0} \frac{q_0qz}{(z^2 + R^2)^{3/2}}\)\)

When \(z \rightarrow +\infty\), \(F_z = \dfrac{1}{4\pi \epsilon_0} \dfrac{q_0q}{z^2}\).

The force between a point charge and a disk of charge is

\[ \begin{aligned} F_z &= \int_0^R \frac{1}{4\pi \epsilon_0} \frac{q_0 (2\pi \omega d \omega \cdot \sigma)}{z^2 + R^2} \frac{z}{\sqrt{z^2 + R^2}} \\ &= \frac{1}{4\pi \epsilon_0} q_0 2\pi\sigma z \int_0^R \frac{\omega d\omega}{(z^2 + \omega^2)^{2/3}} \\ &= \frac{1}{4\pi \epsilon_0} \cdot q_0 \frac{2\pi R^2 \sigma}{R^2} \cdot \left. \frac{z}{\sqrt{z^2 + \omega^2}} \right|_R^0 \\ &= \frac{1}{4\pi \epsilon_0} \frac{2q_0q}{R^2} \left( 1 - \frac{z}{\sqrt{z^2 + R^2}} \right) \\ &= \frac{\sigma q_0}{2\epsilon_0} \left( 1 - \frac{1}{\sqrt{1 + (R/z)^2}} \right) \end{aligned} \]

When \(R \rightarrow +\infty\), \(F_z = \dfrac{\sigma q_0}{2\epsilon_0}\).

Electric Field¶

The electric field generated by an infinite line of charge is

\[ \begin{gathered} dE = \frac{1}{4\pi \epsilon_0} \frac{\lambda dx}{(r / \cos\theta)^2} \\\\ x = r \tan\theta \Rightarrow dx = r \sec^2 \theta d\theta \\\\ E_y = \int_{-\pi/2}^{\pi/2} \frac{1}{4\pi \epsilon_0} \frac{\lambda d\theta}{r} \cos\theta = \frac{\lambda}{2\pi \epsilon_0 r} \end{gathered} \]

The electric field generated by an electric dipole is

\[ \begin{gathered} E_x(r, 0) = 0 \\\\ E_y(r, 0) = -\frac{1}{4\pi \epsilon_0} \frac{2Qa}{(r^2 + a^2)^{3/2}} \\\\ E_x(0, r) = 0 \\\\ E_y(0, r) = \frac{Q}{4\pi \epsilon_0} \left[ \frac{1}{(r-a)^2} - \frac{1}{(r+a)^2} \right] = \frac{1}{4\pi \epsilon_0} \frac{4Qa}{r^3 (1 - a^2/r^2)^2} \\ \end{gathered} \]

We can set \(p = 2Qa = Ql\), where \(l\) is the distance between the two charges. When \(r >> a\), we have \(E_y(r, 0) = -\dfrac{1}{4\pi \epsilon_0} \dfrac{p}{r^3}, E_y(0, r) = \dfrac{1}{4\pi \epsilon_0} \dfrac{2p}{r^3}\)

The torque of a dipole in a uniform electric field is \(\tau = p \times E\).

The work done by the electric dipole moment in a uniform electric field is \(W = \int_{\theta_0}^{\theta} \tau d\theta = -\int_{\theta_0}^{\theta} pE\sin\theta d\theta = pE(\cos\theta - \cos\theta_0)\). So \(U = - \vec{p} \bullet \vec{E}\).

Gauss Law¶

The Gauss Law is: \(\(\Phi_E = \oiint E \cdot dA = \frac{q}{\epsilon_0}\)\)

By this, we can get the electric field inside sphere:

\[ \begin{gathered} q = \frac{4}{3}\pi r^3\rho \\\\ E = \frac{1}{4\pi\epsilon_0} \frac{q}{r^2} = \frac{\rho r}{3\epsilon_0} = \frac{Qr}{4\pi\epsilon_0R^3} \end{gathered} \]

The outer electric field for a charged conductor is:

\[ \begin{gathered} \epsilon_0 \oiint E \cdot dA = \epsilon_0 E \Delta A = \sigma \cdot \Delta A \\\\ \Rightarrow E = \frac{\sigma}{\epsilon_0} \end{gathered} \]

The electric field for a infinite sheet of charge is: \(\(E = \frac{\sigma}{2\epsilon_0}\)\)

The electric field for a infinite line of charge is: \(\(E = \frac{\lambda \cdot l}{\epsilon_0 \cdot 2\pi rl} = \frac{\lambda}{2\pi\epsilon_0 r}\)\)

By the Gauss Law, we can get \(\(\rho = \epsilon_0 \nabla \cdot \vec{E}\)\)

In spherical coordinates, \(\(\nabla \cdot \vec{E} = \frac{1}{r^2} \frac{\partial r^2 E_r}{\partial r} + \frac{1}{r\sin\theta} \frac{\partial \sin\theta E_{\theta}}{\partial \theta} + \frac{1}{r\sin\theta} \frac{\partial E_{\phi}}{\partial \phi}\)\)

In cylindrical coordinates, \(\(\nabla \cdot \vec{E} = \frac{1}{r} \frac{\partial r E_r}{\partial r} + \frac{1}{r} \frac{\partial E_{\theta}}{\partial \theta} + \frac{\partial E_{z}}{\partial z}\)\)

Shorthand¶

Force between a point charge and a ring of charge
Force between a point charge and a disk of charge
Electric field generated by an infinite plane of charge
Electric field generated by an infinite line of charge
Electric field generated by an electric dipole
Electric field inside sphere (Gauss Law)
Electric field for a charged conductor (Gauss Law)
Torque of a dipole in a uniform electric field
Energy of a dipole in a uniform electric field
Volume charge density from electric field