3.S: Electric Potential (Summary)
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Key Terms
| electric dipole | system of two equal but opposite charges a fixed distance apart |
| electric dipole moment | quantity defined as \(\displaystyle \vec{p}=q\vec{d}\) for all dipoles, where the vector points from the negative to positive charge |
| electric potential | potential energy per unit charge |
| electric potential difference | the change in potential energy of a charge q moved between two points, divided by the charge. |
| electric potential energy | potential energy stored in a system of charged objects due to the charges |
| electron-volt | energy given to a fundamental charge accelerated through a potential difference of one volt |
| electrostatic precipitators | filters that apply charges to particles in the air, then attract those charges to a filter, removing them from the airstream |
| equipotential line | two-dimensional representation of an equipotential surface |
| equipotential surface | surface (usually in three dimensions) on which all points are at the same potential |
| grounding | process of attaching a conductor to the earth to ensure that there is no potential difference between it and Earth |
| ink jet printer | small ink droplets sprayed with an electric charge are controlled by electrostatic plates to create images on paper |
| photoconductor | substance that is an insulator until it is exposed to light, when it becomes a conductor |
| Van de Graaff generator | machine that produces a large amount of excess charge, used for experiments with high voltage |
| voltage | change in potential energy of a charge moved from one point to another, divided by the charge; units of potential difference are joules per coulomb, known as volt |
| xerography | dry copying process based on electrostatics |
Key Equations
| Potential energy of a two-charge system | \(\displaystyle U(r)=k\frac{qQ}{r}\) |
| Work done to assemble a system of charges | \(\displaystyle W_{12⋯N}=\frac{k}{2}\sum_i^N\sum_j^N\frac{q_iq_j}{r_{ij}}\) for \(\displaystyle i≠j\) |
| Potential difference | \(\displaystyle ΔV=\frac{ΔU}{q}\) or \(\displaystyle ΔU=qΔV\) |
| Electric potential | \(\displaystyle V=\frac{U}{q}=−∫^P_R\vec{E}⋅\vec{dl}\) |
| Potential difference between two points | \(\displaystyle ΔV_{AB}=V_B−V_A=−∫^B_A\vec{E}⋅\vec{dl}\) |
| Electric potential of a point charge | \(\displaystyle V=\frac{kq}{r}\) |
| Electric potential of a system of point charges | \(\displaystyle V_P=k\sum_1^N\frac{q_i}{r_i}\) |
| Electric dipole moment | \(\displaystyle \vec{p}=q\vec{d}\) |
| Electric potential due to a dipole | \(\displaystyle V_P=k\frac{\vec{p}⋅\hat{r}}{r^2}\) |
| Electric potential of a continuous charge distribution | \(\displaystyle V_P=k∫\frac{dq}{r}\) |
| Electric field components | \(\displaystyle E_x=−\frac{∂V}{∂x},E_y=−\frac{∂V}{∂y},E_z=−\frac{∂V}{∂z}\) |
| Del operator in Cartesian coordinates | \(\displaystyle \vec{∇}=\hat{i}\frac{∂}{∂x}+\hat{j}\frac{∂}{∂y}+\hat{k}\frac{∂}{∂z}\) |
| Electric field as gradient of potential | \(\displaystyle \vec{E}=−\vec{∇}V\) |
| Del operator in cylindrical coordinates | \(\displaystyle \vec{∇}=\hat{r}\frac{∂}{∂r}+\hat{φ}\frac{1}{r}\frac{∂}{∂φ}+\hat{z}\frac{∂}{∂z}\) |
| Del operator in spherical coordinates | \(\displaystyle \vec{∇}=\hat{r}\frac{∂}{∂r}+\hat{θ}\frac{1}{r}\frac{∂}{∂θ}+\hat{φ}\frac{1}{r\sin\theta}\frac{∂}{∂φ}\) |
Summary
7.2 Electric Potential Energy
- The work done to move a charge from point \(A\) to \(B\) in an electric field is path independent, and the work around a closed path is zero. Therefore, the electric field and electric force are conservative.
- We can define an electric potential energy, which between point charges is \(\displaystyle U(r)=k\frac{qQ}{r}\), with the zero reference taken to be at infinity.
- The superposition principle holds for electric potential energy; the potential energy of a system of multiple charges is the sum of the potential energies of the individual pairs.
7.3 Electric Potential and Potential Difference
- Electric potential is potential energy per unit charge.
- The potential difference between points \(A\) and \(B\), \(\displaystyle V_B−V_A\), that is, the change in potential of a charge q moved from \(A\) to \(B\), is equal to the change in potential energy divided by the charge.
- Potential difference is commonly called voltage, represented by the symbol \(\displaystyle ΔV\):
\(\displaystyle ΔV=\frac{ΔU}{q}\) or \(\displaystyle ΔU=qΔV.\)
- An electron-volt is the energy given to a fundamental charge accelerated through a potential difference of 1 V. In equation form,
\(\displaystyle 1\,\mathrm{eV}=(1.60×10^{−19}\,\mathrm{C})(1\,\mathrm{V})=(1.60×10^{−19}\,C)(1\,\mathrm{J}/\mathrm{C})=1.60×10^{−19}\,\mathrm{J}\).
7.4 Calculations of Electric Potential
- Electric potential is a scalar whereas electric field is a vector.
- Addition of voltages as numbers gives the voltage due to a combination of point charges, allowing us to use the principle of superposition: \(\displaystyle V_P=k\sum_1^N\frac{q_i}{r_i}\).
- An electric dipole consists of two equal and opposite charges a fixed distance apart, with a dipole moment \(\displaystyle \vec{p}=q\vec{d}\).
- Continuous charge distributions may be calculated with \(\displaystyle V_P=k∫\frac{dq}{r}\).
7.5 Determining Field from Potential
- Just as we may integrate over the electric field to calculate the potential, we may take the derivative of the potential to calculate the electric field.
- This may be done for individual components of the electric field, or we may calculate the entire electric field vector with the gradient operator.
7.6 Equipotential Surfaces and Conductors
- An equipotential surface is the collection of points in space that are all at the same potential. Equipotential lines are the two-dimensional representation of equipotential surfaces.
- Equipotential surfaces are always perpendicular to electric field lines.
- Conductors in static equilibrium are equipotential surfaces.
- Topographic maps may be thought of as showing gravitational equipotential lines.
7.7 Applications of Electrostatics
- Electrostatics is the study of electric fields in static equilibrium.
- In addition to research using equipment such as a Van de Graaff generator, many practical applications of electrostatics exist, including photocopiers, laser printers, ink jet printers, and electrostatic air filters.