doPhysics · Electromagnetism 1
Electric Potential, Energy and the Dipole
The energy stored by charges and the potential, energy per coulomb, attached to every point of space. Potentials of point charges and charged bodies, the field as minus the slope of the potential, equipotential surfaces, and the electric dipole with its field, its torque and its energy.
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Why this chapter
Why this chapter exists
The field of the last chapter is a vector, and adding vectors from many charges is slow work. The Coulomb force is conservative, so charges also store energy, and energy is a plain number. Divided by the charge, that energy becomes the potential, one number at every point. Potentials add without components, and the field follows from them as a slope. This chapter builds that shortcut. It ends with the electric dipole, the pair of opposite charges hidden inside every polar molecule.
You will learn
By the end of this chapter you will
- Find the potential energy of a pair of charges and of a system of charges, and the self-energy of a shell and a sphere.
- Use energy conservation to find speeds and distances of closest approach of charged particles.
- Define potential and potential difference, and work in volts and electron volts with the right signs of work.
- Find the potential of point charges, rings, arcs, shells, rods and discs.
- Get the field from the potential, \(E = -dV/dx\) and \(\vec E = -\nabla V\), and read V-x and E-x graphs.
- Describe equipotential surfaces and use them to find work and fields.
- Find the field and potential of a dipole on its axis, on its equator and at any point.
- Find the torque, energy and oscillation of a dipole in a uniform field, and the force on it in a non-uniform field.
Before you begin
Before you begin
The chapter on Electric Charge, Coulomb's Law and the Electric Field comes first. From it you should know Coulomb's law, the field \(\vec E = \vec F/q\) and the fields of point charges, rings and discs. From Mechanics you need potential energy and conservative forces, \(F = -dU/dx\), the torque of a couple, and simple harmonic motion. You should be able to differentiate and to integrate \(1/x\) and \(1/x^2\).
Lessons
1
Electric Potential Energy
The Coulomb force is conservative, so a pair of charges stores energy in its separation, \(U = kq_1q_2/r\) with zero at infinity. Its sign tells whether the pair wants to fly apart or to stay together, and released charges turn it into motion.
Work in a field and potential energyWork to change a separationEnergy becomes motion
2
Electric Potential
Potential is potential energy per unit charge, a number attached to every point of space. For a point charge it is kq/r, potentials of many charges simply add, and a potential difference measures the work per coulomb, in volts or in electron volts.
Potential as energy per unit chargePotential of a point chargePotentials add as numbersPotential difference and the electron volt
3
Potential of Bodies and Energy of Systems
A charged body is a sum of small charges, so its potential is a sum of kdq/r: easy for a ring or a shell, an integral for a rod or a disc. The energy of many charges is a sum over pairs, and a shell or a sphere carries a self-energy of its own.
Potential of rings shells rods and discsEnergy of a system of chargesSelf energy of a shell and a sphere
4
Field from Potential and Equipotential Surfaces
The field is minus the slope of the potential, \(E = -dV/dx\), so it points where \(V\) falls fastest. In a uniform field \(V\) falls by \(Ed\) along the field. Surfaces of equal potential cross the field at right angles, and moving a charge along one takes no work.
Field is minus the potential gradientReading potential and field graphsUniform field and potential differenceEquipotential surfaces and work
5
The Electric Dipole
Two equal and opposite charges a small distance apart. Their moment \(p = q \times 2a\) points from \(-q\) to \(+q\). Far away their fields nearly cancel, so the field falls as \(1/r^3\): \(2kp/r^3\) on the axis, \(kp/r^3\) on the equator, and a potential \(kp\cos\theta/r^2\).
The dipole momentAxial and equatorial fieldPotential of a dipoleField at any point near a dipole
6
The Dipole in an Electric Field
A uniform field puts a couple on a dipole, \(\tau = pE\sin\theta\), with no net force. Its energy \(U = -pE\cos\theta\) is lowest when \(p\) lies along \(E\), so that is the stable position, and a dipole nudged from it oscillates. A non-uniform field also pulls: an aligned dipole moves into the stronger field.
Torque on a dipolePotential energy of a dipoleSmall oscillations of a dipoleA dipole in a non uniform field
Read in order. The small labels are the topics of each lesson. They take your colour as you answer the checks.
0%The Workbook
Questions to solve on paper, marked by you.
Glossary
The key words of this chapter, with the meaning this chapter gives them.
- axial line
- The line through the two charges of a dipole. On it the field of a short dipole is \(2kp/r^3\), along \(\vec p\).
- dipole moment
- The vector \(\vec p = q \cdot 2\vec a\) of a dipole, of size \(2qa\), pointing from \(-q\) to \(+q\); unit C m.
- distance of closest approach
- The nearest distance a charged particle fired at a like charge reaches. There all its kinetic energy has become potential energy, and it stops for an instant.
- electric dipole
- A pair of equal and opposite charges \(+q\) and \(-q\) a small distance \(2a\) apart.
- electric potential
- The work per unit charge done by an outside agent in bringing a small positive charge slowly from infinity to a point, \(V = W/q_0\). A scalar that belongs to a point in space.
- electric potential energy
- Energy stored in an arrangement of charges. For two point charges it is \(U = kq_1q_2/r\), zero when they are infinitely far apart, positive for like charges and negative for unlike ones.
- electron volt
- The energy gained by a particle of charge \(e\) crossing a potential difference of one volt: \(1\,\text{eV} = 1.6 \times 10^{-19}\,\text{J}\).
- equatorial line
- The perpendicular bisector of a dipole. On it the potential is zero and the field of a short dipole is \(kp/r^3\), against \(\vec p\).
- equipotential surface
- A surface on which every point has the same potential. The field is perpendicular to it, and no work is done in moving a charge along it.
- polar molecule
- A molecule whose positive and negative charges are permanently a little apart, so that it has a dipole moment, like water or HCl.
- potential difference
- The difference of potential between two points, \(V_B - V_A\). The work to move a charge \(q\) slowly from A to B is \(q(V_B - V_A)\).
- potential gradient
- The rate of change of potential with distance, \(dV/dx\) along a line or \(\nabla V\) in space. The electric field is its negative.
- self-energy
- The energy needed to gather the charge of a single body from infinity. It is \(kQ^2/2R\) for a thin shell and \(3kQ^2/5R\) for a uniformly charged solid sphere.
- short dipole
- A dipole seen from distances much larger than the separation of its charges, where its field is \(\propto p/r^3\) and its potential \(kp\cos\theta/r^2\).
- volt
- The unit of potential and potential difference: \(1\,\text{V} = 1\,\text{J/C}\).