Basic Laws - Coulomb, Ampere, Faraday

Coulomb's Law - The force between to charges is inversely proportional to the square of separating distance, r², and the direction is along the vector from one to the other. The force on q₁ due to q along a vector, $\mathbf{r}$, from q to q₁ is listed below.

$$\mathbf{F} = \frac{1}{4 \pi \epsilon_0} \frac{q_1 q}{r^2} \m... ...athbf{r} =\frac{-q_1 q}{4 \pi \epsilon_0} \nabla \; \frac{1}{r}$$ (7)

The electrostatic field is simple the potential force on an arbitrary charge and derived from $\mathbf{F} / q_1$. Both fields are simple additive with multiple sources of charge.

$$\mathbf{E} = \frac{1}{4 \pi \epsilon_0} \frac{q}{r^2} \mathb... ...} \mathbf{r} =\frac{-q}{4 \pi \epsilon_0} \nabla \; \frac{1}{r}$$ (8)

Gauss' Law of Electricity - One can think of the electric field as a flux with positive and negative charges acting as source and sinks, respectively. Integrating this flux over a closed surface and relating it to the source and sinks is done with Gauss' Law.

$$\iint_{A} \mathbf{E} \cdot \mathbf{\hat{n}} \; dA = \frac{q}{\epsilon_0}$$ (9)

Using the divergence theorem and the concept of charge density, $q=\iiint_{V} \rho_e \; dV$, we can strip the integrals away and produce a differential form of Gauss' Law.

$$\nabla \cdot \mathbf{E} = \frac{\rho_e}{\epsilon_0}$$ (10)

Biot-Savart Law - The magnetic field induced by a moving charge where q is the charge and $\mathbf{V}_q$ is it's velocity. Like the electric field, this relationship is additive for multiple moving charges or currents.

$$\mathbf{B} = \frac{\mu_0}{4 \pi} \frac{q\mathbf{V}_q \times... ...ac{\mu_0}{4 \pi} \; q\mathbf{V}_q \times \nabla \; \frac{1}{r}$$ (11)

$$d\mathbf{B} = \frac{\mu_0}{4 \pi} \frac{dq\mathbf{V}_q \tim... ..._0}{4 \pi} \frac{i \; d\mathbf{s} \times \mathbf{\hat{r}}}{r^2}$$ (12)

Ampere's Law - Just as positive and negative charges act as sources and sinks of an electric field, a moving charges acts like a vortex creating a magnetic field. The circulation of the magnetic field around a closed loop is proportional to the quantity of moving charge or current passing through the loop.

$$\oint_{C} \mathbf{B} \cdot \mathbf{\hat{t}} \; ds = \mu_0 i$$ (13)

Using Stoke's Law and expanding the value of i into an integral will produce a differential form with the intergrals dropped.

$$\nabla \times \mathbf{B} = \mu_0 \left( \mathbf{J} + \epsilon_0 \frac{\partial \mathbf{E}}{\partial t} \right)$$ (14)

Gauss' Law of Magnetism - This is the exact same concept of Gauss' law as applied in the electric field case with respect to magnetic fields. There must be conservation of magnetic flux line through a closed surface. However, unlike electric fields there are no magnetic sources and sinks. Therefore, the net flux must equal zero. The differential form listed follows from the divergence theorem.

$$\iint_{A} \mathbf{B} \cdot \mathbf{\hat{n}} \; dA = 0$$ (15)

$$\nabla \cdot \mathbf{B} = 0$$ (16)

Faraday's Law of Induction - A changing magnetic flux through a surface capping a closed loop produces a proportional induced electric field. An induced electric field differs from a normal one in the fact that all fields lines form closed loops rather than connect positive and negative charges. The concept of electric potential has little meaning with respect to induced electric fields.

$$\oint_{C} \mathbf{E} \cdot \mathbf{\hat{t}} \; ds = - \frac{d}{dt} \iint_A \mathbf{B} \cdot \mathbf{\hat{n}} \; dA$$ (17)

Using Stoke's Theorem and dropping the surface integrals we get Faraday's Law in differential form.

$$\nabla \times \mathbf{E} = \frac{\partial \mathbf{B}}{\partial t}$$ (18)