16.2 Electrical potential energy

Electrical potential energy and potential

The electrical potential energy of a charge is the energy it has because of its position relative to other charges that itinteracts with. The potential energy of a charge $Q_1$ relative to a charge $Q_2$ a distance $r$ away is calculated by:

Electrical potential

The electrical potential at a point is the electrical potential energy per unit charge, i.e. the potential energy a positive testcharge would have if it were placed at that point.

Consider a positive test charge $+Q$ placed at A in the electric field of another positive point charge.

The test charge moves towards B under the influence of the electric field of the other charge.In the process the test charge loses electrical potential energy and gains kinetic energy. Thus, at A, the test charge has morepotential energy than at B – A is said to have a higher electrical potential than B .

The potential energy of a charge at a point in a field is defined as the work required to move thatcharge from infinity to that point.

Potential difference

The potential difference between two points in an electric field is defined as the work required to move a unit positive test charge from the point of lower potential to that of higherpotential .

If an amount of work $W$ is required to move a charge $Q$ from one point to another, then the potential difference between the two points is given by,

From this equation we can define the volt.

The Volt

One volt is the potential difference between two points in an electric field if one joule ofwork is done in moving one coulomb of charge from the one point to the other.

Real-world application: lightning

Lightning is an atmospheric discharge of electricity, usually, but not always, during a rain storm. An understanding of lightning isimportant for power transmission lines as engineers need to know about lightning in order to adequately protect lines andequipment.

Formation of lightning

1. Charge separation The first process in the generation of lightning is charge separation. The mechanism by which charge separation happens is still the subject of research.One theory is that opposite charges are driven apart and energy is stored in the electric field between them. Cloud electrification appears to require strongupdrafts which carry water droplets upward, supercooling them to $-10$ to $-20$   ${}^{\circ }$ C. These collide with ice crystals to form a soft ice-water mixture called graupel. The collisions result in a slight positive charge beingtransferred to ice crystals, and a slight negative charge to the graupel. Updrafts drive lighter ice crystals upwards, causing the cloud top to accumulateincreasing positive charge. The heavier negatively charged graupel falls towards the middle and lower portions of the cloud, building up an increasing negativecharge. Charge separation and accumulation continue until the electrical potential becomes sufficient to initiate lightning discharges, which occurs whenthe gathering of positive and negative charges forms a sufficiently strong electric field.
2. Leader formation As a thundercloud moves over the Earth's surface, an equal but opposite charge is induced in the Earth below, and the induced ground charge follows themovement of the cloud. An initial bipolar discharge, or path of ionized air, starts from a negativelycharged mixed water and ice region in the thundercloud. The discharge ionized channels are called leaders. The negative charged leaders, called a "steppedleader", proceed generally downward in a number of quick jumps, each up to 50 metres long. Along the way, the stepped leader may branch into a number of pathsas it continues to descend. The progression of stepped leaders takes a comparatively long time (hundreds of milliseconds) to approach the ground. Thisinitial phase involves a relatively small electric current (tens or hundreds of amperes), and the leader is almost invisible compared to the subsequentlightning channel. When a step leader approaches the ground, the presence of opposite charges onthe ground enhances the electric field. The electric field is highest on trees and tall buildings. If the electric field is strong enough, a conductivedischarge (called a positive streamer) can develop from these points. As the field increases, the positive streamer may evolve into a hotter, higher currentleader which eventually connects to the descending stepped leader from the cloud. It is also possible for many streamers to develop from many differentobjects at the same time, with only one connecting with the leader and forming the main discharge path. Photographs have been taken on which non-connectedstreamers are clearly visible. When the two leaders meet, the electric current greatly increases. The region of high current propagates back up the positivestepped leader into the cloud with a "return stroke" that is the most luminous part of the lightning discharge.
3. Discharge When the electric field becomes strong enough, an electrical discharge (the bolt of lightning) occurs within clouds or between clouds and the ground. During thestrike, successive portions of air become a conductive discharge channel as the electrons and positive ions of air molecules are pulled away from each other andforced to flow in opposite directions. The electrical discharge rapidly superheats the discharge channel, causing theair to expand rapidly and produce a shock wave heard as thunder. The rolling and gradually dissipating rumble of thunder is caused by the time delay of soundcoming from different portions of a long stroke.

Estimating distance of a lightning strike. The flash of a lightning strike and resulting thunder occur at roughly the sametime. But light travels at 300 000 kilometres in a second, almost a million times the speed of sound. Sound travels at the slowerspeed of 330 m/s in the same time, so the flash of lightning is seen before thunder is heard. By counting the seconds between theflash and the thunder and dividing by 3, you can estimate your distance from the strike and initially the actual storm cell (inkilometres).