Introduction

Recently I am preparing for a exam whcih referes to the AP E&M exam so I deceide to make such file to track my learnings.

Learning route

I started with reading Introduction to Electrodynamics by David J. Griffith, however, that is pretty tough to consider the real exact math work, so I decided to just understand the brief part and turn to problems directly.

Problems and my Comprehensions

1. Two negative poit charges, both of magnitude $6.0 \times 10^{-6} C$, are situated along the x-axis at $x_1 = -2.0 m$ and $x_2 = + 2.0 m$, what is the electric potential at the origin?

Most of the time E vanishes at infinite, so define it from infinite point. By the curl-less-ness of E, we have the integral actually is releventless of the route it tracks.

So we can calculate in this way:

And because the potential is a scalar, so direction doesn’t matters final value shoule be $kq = 9 \times 10^{9} \times 6 \times 10 ^{-6} = 5.4 \times 10^{4}$

$k \approx 8.99 \times 10^9\text{ N}\cdot\text{m}^2/\text{C}^2$ (commonly rounded to $9.0 \times 10^9\text{ N}\cdot\text{m}^2/\text{C}^2$ for standard multiple-choice problems).

In a shallower version, we can just apply $V = \frac{kq}{r}$ which all of the variables are scalars.

2. An electric dipole consisting of a positive charge and a negative charge held a fixed distance apart is at rest in an external, nonuniform electric field E, as shown in the figure above. Which of the following best describes the net torque and net force exerted on the dipole? Check the direction of the net Torque and Net Force.

3. In the figure is a solid, isolated, metallic conductor in electrostatic equilibrium with a net charge +Q. X and Y are at each end on the conductor. Compare the Potential on X and Y.

That is absolutely Wrong.

4. About capacitor

• Define Capacitance: $C=\dfrac{Q}{V}$.
  • However Cpacitance is defined by the quotient, but factors determined is on the properties of the capacitor itself.
  • For a standard parallel-plate capacitor, capacitance is determined by the formula:$$C = \frac{\varepsilon_0 A}{d}$$ Which $d$ is the distances, $A$ is the surface area.
  • The Material Factor (Dielectrics):$$C = \frac{\kappa \varepsilon_0 A}{d}$$, $\kappa$: dielectric constant
• Energy stored: $$U = \int_{0}^Q \frac{q}{C}dq = \frac{1}{2}\frac{Q^2}{C} =\frac{1}{2}QV$$
• in one circuit: From the rule of how capacitors store charges. When it has already reached steady state, we can derive that $Q_{C_1}=Q_{C_{2,3equally}}$ and when we consider 2,3, $V_{C_2}=V_{C_3}$, so ${C_{2,3}}={C_2}+{C_3}$

5. A long, straight wire of radius R carries current I. The current is distributed over the cross-sectional area of the wire with a uniform current density. Which of the following graphs best represents the magnetic field strength produced by the current as a function of the distance r from the center of the wire?

So actually, when $r<R$ at a certain point inside a cylindral, we just need to consider the cycle contains it, it is like from cylindral to a circle pancake to a circle linear which radii $r$.

we have $B(2\pi r)= \mu_0I\cdot\frac{r^2}{R^2}$ then $\mathbf{B}$ is proportional to radius $r$.

Especially, point outside of a line current, we can see it as $B(2\pi r) = \mu_0 I \Rightarrow B = \frac{\mu_0I}{2\pi}$