# 23.1 Electric Potential Energy and Difference #

- PE can only be defined for conservative forces
- That is, work done by said force is independent of the path taken
- Coulomb’s Law is conservative because the dependence on position is conservative

- Hence, we define .$\Delta U = -W$ with
- .$\Delta U = U_b - U_a$ is for a situation where a point charge .$q$ moves from point .$a$ to point .$b$
- This is equal to negative work, .$-W = -\vec F d = -(q\vec E) d$ (for a uniform .$\vec E$)

# 23.2 Relation between Electric Potential and Field #

**Electric Potential:**Electric PE per unit charge, such as for a charge at point .$a$ $$V_a = \frac{U_a}{q}$$- We only really care about difference though, which is defined as $$V_{ba} = \Delta V = \frac{U_b - U_a}{q} = - \frac{W_{ba}}{q}$$
- We can now also define PE in terms of electric potential: $$\Delta U = U_b - U_a = q(V_b - V_a) = qV_{ba}$$
- Electric potential difference is a measure of how much energy an electric charge can acquire in a given situation.
- Since energy is the ability to do work, the electric potential difference is also a measure of how much work a given charge can do.
- The exact amount of energy or work depends both on the
*potential difference*and on*the charge*.

- The exact amount of energy or work depends both on the
- If a positive charge is free, it will tend to move from high to low potential
- Inverse for opposite charge

# 23.3 Potential due to Point Charges #

$$\Delta U = U_b - U_a = - \int_a^b \vec F \cdot d \vec l$$

- .$dl$ is an infinitesimal increment of displacement along the path from .$a$ to .$b$
- Keep in mind that .$\vec F$ must be conservative
- Thus the integral can be taken along any path from point .$a$ to point .$b$.

- Knowing .$\vec E = \vec F / q$ and .$V_{ba} = (U_b - U_a) / q$, we can write the electric potential equation as…
$$V_{ba} = V_b - V_a = - \int_a^b \vec E \cdot d \vec l$$ $$V_{ba, \text{uniform $\vec E$}} = -E\int_a^b d\vec l = -Ed$$ …where .$d$ is the distance of a straight line from point .$a$ to .$b$

## Charged Conducting Sphere #

### 1. Electric Potential **Outside** Sphere
#

- We know .$\vec E = \frac{1}{4\pi\varepsilon_0} \frac{Q}{r^2}$ for outside a conducting sphere (.$r > r_0$)
- Therefore, we can write $$V_{ba} = - \int_{r_a}^{r_b} \vec E \cdot d \vec l = - \frac{Q}{4\pi\varepsilon_0}\int_{r_a}^{r_b} \frac{dr}{r^2}$$ $$\dots = \frac{Q}{4\pi\varepsilon_0} \bigg(\frac{1}{r_b} - \frac{1}{r_a}\bigg)$$ $$\dots = \frac{1}{4\pi\varepsilon_0} \frac{Q}{r} \text{ [$r_b = \infty$]}$$

### 2. Electric Potential **On** Sphere
#

- From .$(a)$, as .$r$ approaches .$r_0$, we see $$V = \frac{1}{4\pi\varepsilon_0} \frac{Q}{r_0}$$ at the surface of the sphere. This makes sense because the charge is distributed on the surface of the sphere.

### 3. Electric Potential **Inside** Sphere
#

- Inside the conductor, .$\vec E = 0$
- Therefore, there is no change in .$\vec E$ from .$0$ to .$r_0$ (or any point within the conductor) gives zero change in .$V$
- Hence, within the conductor, .$V$ is a constant: $$V = \frac{1}{4\pi\varepsilon_0} \frac{Q}{r_0}$$

- Thus, the whole conductor, not just its surface, is at this same potential.
- We can also generalize the first case to the electric potential .$r$ from a
**single point charge**.$Q$

## Coulomb potential #

- The potential outside a uniformly charged sphere is the same as if all the charge were concentrated at its center
The potential near a positive charge is large, and it decreases toward zero at very large distances

For a negative charge, the potential is negative and increases toward zero at large distances

# 23.4 Potential due to Any Charge Distribution #

- If .$\vec E$ is a function of position (or otherwise unknown), we can find .$V$ by calculating the potential due to the many tiny charges that make up .$\vec E$: $$V = \frac{1}{4\pi\varepsilon_0} \int \frac{dq}{r}$$ where .$r$ is the distance from a tiny element of charge .$dq$ to the point where .$V$ is being determined

# 23.5 Equipotential Lines and Surfaces #

- The electric potential can be represented by drawing
**equipotential lines**, or, in three dimensions,**equipotential surfaces** - An equipotential surface has all points at the same potential.
- That is, the potential difference between any two points on the surface is zero
- Thus, no work is required to move a charge from one point on the surface to another.

- Equipotential surfaces are perpendicular to the electric field (field lines)
- For a
*positive*point charge, the equipotential surface with the largest potential is closest to the*positive*charge - Unlike electric field lines, which start and end on electric charges, equipotential lines/surfaces are always continuous and never end

Electric field lines and equipotential surfaces for a point charge.

Equipotential lines (green, dashed) are always perpendicular to the electric field lines (solid red) shown here for two equal but oppositely charged particles (an electric dipole).

# 23.6 Potential Due to Dipole (Moment) #

$$V = \frac{1}{4\pi\varepsilon_0} \frac{Q}{r} + \frac{1}{4\pi\varepsilon_0} \frac{(-Q)}{(r+\Delta r)} = \frac{Q}{4\pi\varepsilon_0} \frac{\Delta r}{r(r + \Delta r)}$$

- .$r$ is the distance from (some arbitrary point) .$P$ to the positive charge and .$r + \Delta r$ is the distance to the negative charge
- If .$r \gg l$, then .$r \gg \Delta r \approx l \cos \theta$ so we can neglect .$\Delta r$ $$V = \frac{1}{4\pi\varepsilon_0} \frac{Ql \cos\theta}{r^2} = \frac{1}{4\pi\varepsilon_0} \frac{p \cos\theta}{r^2} $$
- Notice the potential decreases .$\propto r^2$, whereas for a single point charge the potential decreases .$\propto r$
- It is not surprising that the potential should fall off faster for a dipole:
- When you are far from a dipole, the two equal but opposite charges appear so close together as to tend to neutralize each other

# 23.7 .$\vec E$ Determined from .$V$ #

- We know that .$V_b - V_a = - \int_a^b \vec E \cdot d\vec l$, which we can write in differential form as .$dV = -\vec E \cdot d\vec l = - E_l dl$. This can be written as $$E_l = - \frac{dV}{dl}$$
- .$dV$ is the tiny difference in potential between two points a distance .$dl$ apart, and .$E_l$ is the component of the electric field in the direction of the tiny displacement .$d\vec l$
- This is called the gradient of .$V$ in a particular direction: The general case is $$\vec E = - \nabla \vec V = - \bigg\langle \frac{\delta V}{\delta x}, \frac{\delta V}{\delta y}, \frac{\delta V}{\delta z} \bigg\rangle$$
- This states that the electric field points “downhill” towards lower voltages (where there is lower potential)

# 23.8 Electrostatic PE; The Electron Volt #

- The electric potential and energy potential due to one point charge .$Q_1$ on another point charge .$Q_2$ separated by .$r_{12}$ are $$V = \frac{1}{4\pi\varepsilon_0} \frac{Q_1}{r_{12}}$$ $$U = Q_2 V = \frac{1}{4\pi\varepsilon_0} \frac{Q_1 Q_2}{r_{12}}$$
- The PE is the negative work needed to separate the two charges to infinity.
- For three points, we can use the superposition principle like we have prior to write $$U = \frac{1}{4\pi\varepsilon_0}\bigg( \frac{Q_1 Q_2}{r_{12}}+ \frac{Q_1 Q_3}{r_{13}} + \frac{Q_2 Q_3}{r_{23}} \bigg)$$

## Electron Volt #

- Joules are a very large unit for dealing with energy of the electron scale; as such, the
**electron volt**(.$eV$) is often used - One electron volt is the energy acquired by a particle carrying a charge .$e$ (the magnitude of an electron) as a result of moving through a potential difference of .$1 V$ $$1 \text{ eV} = 1.6022 \cdot 10^{-19} \text{ J}$$
- E.x., an electron (charge .$e = 1.6\cdot10^{-19}$) that accelerates through a potential difference of .$1000 \text{ V}$ will lose .$1000 \text{ eV}$ of potential energy and gain .$1000 \text{ eV}$ of kinetic energy

# 23.9 Digital; Binary Numbers; Signal Voltage (not covered) #

- Batteries and wall sockets provide a steady
**supply voltage**as*power* **Signal voltage**provide/carry*information***Analog**signal voltage has voltage that varies continuously (i.e .$\sin$)**Digital**signals are more complicated and encode information, often in binary- Bytes have 8 bits which allow .$2^8 = 256$ numbers
- Digital signals are transmitted at some rate (bit-rate) given in .$\text{Mb/s}$

- Analog to digital converters, ADCs, convert analog signals to boxy digital waves
- The difference between the original continuous and it’s digital approximation is called the
**quantization error / loss** - This error varies by primarily:
**Resolution**or**bit depth**which is the number of bits for the voltage of each sample**Sampling rate**which is the number of times per second the original analog voltage is measured (sampled)

- E.x., CDs are sampled at .$44.1 \text{ kHz}$ with a bit depth of .$16 \text{ bits per sample}$

- The difference between the original continuous and it’s digital approximation is called the

The red analog sine wave, which is at a 100-Hz frequency (1 wavelength is done in 0.010 s), has been converted to a 2-bit (4 level) digital signal (blue).

- Digital Signals
- Digital to Analog, DACs, exist too because some appliances require an analog signal
- Digital signals can be compressed: Repeated information can be reduced so that less memory (bits) is needed
- Fun fact: Bit is the contraction of “
**b**inary dig**it**”, leaving out the 8 letters between

- Fun fact: Bit is the contraction of “
- Digital signals are more resistant from noise, which badly corrupts analog signals
- Any electronic signal involves electric charges whose electric field can affect charges in another nearby signal
- External fields, as from high voltage wires, motors, or fluorescent lamps, can produce noise
- Thermal noise refers to random motion of electrons, much like the â€śthermal motionâ€ť of the molecules in a gas
- Moving electrons can be affected by the medium (wire, etc.), altering the signal