7.8: Electric Potential (Summary)
( \newcommand{\kernel}{\mathrm{null}\,}\)
Key Terms
electric dipole | system of two equal but opposite charges a fixed distance apart |
electric dipole moment | quantity defined as →p=q→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 | U(r)=kqQr |
Work done to assemble a system of charges | W12⋯N=k2N∑iN∑jqiqjrij for i≠j |
Potential difference | ΔV=ΔUq or ΔU=qΔV |
Electric potential | V=Uq=−∫PR→E⋅→dl |
Potential difference between two points | ΔVAB=VB−VA=−∫BA→E⋅→dl |
Electric potential of a point charge | V=kqr |
Electric potential of a system of point charges | VP=kN∑1qiri |
Electric dipole moment | →p=q→d |
Electric potential due to a dipole | VP=k→p⋅ˆrr2 |
Electric potential of a continuous charge distribution | VP=k∫dqr |
Electric field components | Ex=−∂V∂x,Ey=−∂V∂y,Ez=−∂V∂z |
Del operator in Cartesian coordinates | →∇=ˆi∂∂x+ˆj∂∂y+ˆk∂∂z |
Electric field as gradient of potential | →E=−→∇V |
Del operator in cylindrical coordinates | →∇=ˆr∂∂r+ˆφ1r∂∂φ+ˆz∂∂z |
Del operator in spherical coordinates | →∇=ˆr∂∂r+ˆθ1r∂∂θ+ˆφ1rsinθ∂∂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 U(r)=kqQr, 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, VB−VA, 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 ΔV:
ΔV=ΔUq or ΔU=qΔV.
- An electron-volt is the energy given to a fundamental charge accelerated through a potential difference of 1 V. In equation form,
1eV=(1.60×10−19C)(1V)=(1.60×10−19C)(1J/C)=1.60×10−19J.
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: VP=kN∑1qiri.
- An electric dipole consists of two equal and opposite charges a fixed distance apart, with a dipole moment →p=q→d.
- Continuous charge distributions may be calculated with VP=k∫dqr.
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.