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Electricity & Magnetism

Electric Potential & Energy

Trade five vectors for five signed numbers and lose nothing at all in the exchange.

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Electric Potential & Energy

Trade five vectors for five signed numbers and lose nothing at all in the exchange.

30 min3 checkpoints4 simulation tasks

Introduction

Five charges are pinned to a table. The force on a sixth means adding five vectors; the energy to bring it in means adding five signed numbers.

Predict

A +2 nC bead slides from A to B on a frictionless rail: straight, 9.00 m, or arched, 10.42 m. How much work does the field do along each?

Only the endpoints count

Try it

On the straight rail, 9.00 m, the field does 19.137 µJ of work on the bead. Send it over Three humps instead and compare.

GoalSend the bead over Three humps, all the way to B.

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The electric force is conservative: its work depends only on where a trip starts and ends. So it can be written with one number at each point, the potential VV:

W=q(VA−VB)W = q(V_A - V_B)

A route along a contour of equal VV costs nothing. A route that crosses contours pays for each one, whatever the scenery.

Adding numbers instead of vectors

Try it

The bead sits at A, where the potential is 4.78 kV. Make its potential energy negative, and watch what happens to the indigo map.

GoalMake the bead's potential energy negative.

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You just met the two numbers. For point charges, the potential at a point is a plain signed sum, and a test charge qq there has energy U=qVU = qV:

V=k∑iqiri,U=qVV = k\sum_i \frac{q_i}{r_i}, \qquad \pot{U} = qV

VV belongs to the source charges and is measured in volts, joules per coulomb. UU belongs to the pairing, and flips sign with qq.

Practice

Two charges of µC each sit m apart. What is the potential midway between them?

Practice

A uniform field of kV/m points from A toward B, 9.00 m away. How much work does it do on a nC bead making that trip?

The right angle is a theorem

Predict

Two charges of +4 µC and −4 µC sit 3 m apart. What is true at the exact midpoint between them?

Try it

At O, the dipole gives V = 0 and a strong field. Change the arrangement until it is the other way round: no field at O, but a potential.

GoalNo field at O, but a potential.

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Potential has no direction, but its differences do. The field points down the steepest slope of VV, across the contours and at right angles to every one:

E⃗=−∇V\vec{E} = -\nabla V

It is strongest where the contours crowd together, and it can be large where VV itself is zero.

The budget, and what the rail is not allowed to do

Predict

"A positive charge always moves toward lower potential." True?

Try it

A positive bead between two positive charges has to climb toward higher V. Launch it fast enough to get over the hill to B.

GoalGet the bead over the hill to B.

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The bars are KK, UU and their sum, and the K+UK + U bar never moves. The electric force is conservative, and already counted in UU. The rail pushes at right angles to the bead's motion, so it does no work: the same reason the normal force never appeared in the skater's energy budget.

Practice

A 2.0 mg bead carrying nC leaves A at m/s. At B the potential is 36 kV lower. What is its kinetic energy there?

What comes next

You sent a bead the long way round for the same work, flipped its energy without touching the map, and paid for a hill in kinetic energy. Next the charges are free to move, through a metal, across a gap, around a loop, and the potential difference stops being bookkeeping and starts being the thing that drives them.

Ideas this lesson took on

Each of these is a common intuition the lesson put to the test. If one still feels right, the section that tests it is worth another look.

  • “The longer route takes more work”
  • “Where the potential is zero, the field is zero too”
  • “Potential is a vector and points somewhere”
  • “A positive charge always moves toward lower potential”

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Research behind this lesson

  • Chabay & Sherwood 2006: Introductory electricity and magnetism defeats students less because the ideas are hard than because they arrive as a rapid parade of abstract formulas; a course rebuilt round conceptual coherence and a single clear story line, field, then matter, then radiation, teaches the same content better.
  • Lawson & McDermott 1987: Students who could state both theorems often could not apply them, and frequently treated energy as a substance that gets used up rather than a quantity that is transferred.
  • Lindsey 2014: The same incorrect reasoning patterns turned up in students' written answers and interviews about electrostatic potential energy after instruction as in their answers about gravitational potential energy in mechanics: evidence that the energy idea was never built robustly enough the first time to survive being carried into a new context.
  • Maloney et al. 2001: The Conceptual Survey of Electricity and Magnetism (CSEM): a 32-question multiple-choice instrument refined over four years and administered to more than 5000 introductory students at 30 institutions. Pretest averages were 31% (calculus-based) and 25% (algebra/trigonometry-based); after instruction they reached only 47% and 44%.
  • Törnkvist et al. 1993: Shown canonical electric-field-line diagrams and asked to give the force on, and the trajectory of, a test charge, students did not hold apart the field line, the force vector, the velocity vector and the trajectory: the single drawn line was read as all four at once.