Simple Nature - Light and Matter

No. Consider, for instance, an alternative universe in which electricforces are twice as strong as in ours. The numerical value ofk is doubled. Because k is doubled, all the electric field strengthsare doubled as well, which quadruples the quantity E 2 . In the expressionE 2 /8πk, we’ve quadrupled something on top and doubledsomething on the bottom, which makes the energy twice asbig. That makes perfect sense.Potential energy of a pair of opposite charges example 16Imagine taking two opposite charges, b, that were initially farapart and allowing them to come together under the influenceof their electrical attraction.According to our old approach, electrical energy is lost becausethe electric force did positive work as it brought the charges together.(This makes sense because as they come together andaccelerate it is their electrical energy that is being lost and convertedto kinetic energy.)By the new method, we must ask how the energy stored in theelectric field has changed. In the region indicated approximatelyby the shading in the figure, the superposing fields of the twocharges undergo partial cancellation because they are in opposingdirections. The energy in the shaded region is reduced bythis effect. In the unshaded region, the fields reinforce, and theenergy is increased.It would be quite a project to do an actual numerical calculation ofthe energy gained and lost in the two regions (this is a case wherethe old method of finding energy gives greater ease of computation),but it is fairly easy to convince oneself that the energy isless when the charges are closer. This is because bringing thecharges together shrinks the high-energy unshaded region andenlarges the low-energy shaded region.A spherical capacitor example 17⊲ A spherical capacitor, c, consists of two concentric spheres ofradii a and b. Find the energy required to charge up the capacitorso that the plates hold charges +q and −q.⊲ On page 102, I proved that for gravitational forces, the interactionof a spherical shell of mass with other masses outside itis the same as if the shell’s mass was concentrated at its center.On the interior of such a shell, the forces cancel out exactly.Since gravity and the electric force both vary as 1/r 2 , the sameproof carries over immediately to electrical forces. The magnitudeof the outward electric field contributed by the charge +q of thecentral sphere is therefore{ 0, r < a|E + | =kq/r 2 ,, r > awhere r is the distance from the center. Similarly, the magnitudeb / Example 16.c / Example B. Part of theoutside sphere has been drawnas if it is transparent, in order toshow the inside sphere.Section 10.4 Energy In Fields 585