Two equal sized charged spheres, sphere 𝐴 and sphere 𝐵, have charges of plus 𝑞 one and negative 𝑞 two respectively. The spheres are separated by a straight-line distance 𝑑. The net electric field produced by the spheres has a null point along 𝑑 at a distance of 𝑑 over four meters from sphere 𝐴. What is the ratio of the magnitude of the charge of sphere 𝐵 to that of sphere 𝐴?
Let’s start off by drawing a diagram of these two spheres. Our scenario involves two charged spheres, sphere 𝐴 with charge positive 𝑞 one and sphere 𝐵 with charge negative 𝑞 two. They are separated by a distance 𝑑. And we’re further told that at a point on this line a distance of 𝑑 over four meters from sphere 𝐴, the electric field is zero.
The statement asked us to solve for the ratio of the magnitude of the charges 𝑞 two and 𝑞 one. We can start figuring that out by recalling the equation for the electric field created by a point charge. A point charge, 𝑞, creates an electric field, 𝐸, equal to the magnitude of the charge times 𝑘, Coulomb’s constant, divided by 𝑟 squared, where 𝑟 is the distance from the charge to the point where the field 𝐸 is being measured.
Our diagram shows that the charges on sphere 𝐴 and 𝐵 create an electric field of zero at a particular location. If we call that field capital 𝐸, then 𝐸 equals 𝑘 times 𝑞 one, the charge on sphere 𝐴, divided by 𝑑 over four quantity squared plus 𝑘 times 𝑞 two, the charge on sphere 𝐵, divided by the distance three 𝑑 over four quantity squared.
This sum, we’re told, is equal to zero. Considering this equation, we see that we can cancel out both the factors of 𝑘 and the factors of 𝑑 over four in our denominators. That leaves us with an equation saying that 𝑞 one plus 𝑞 two over three squared, or nine, is equal to zero or that 𝑞 two equals negative nine 𝑞 one.
Since our problem statement specifically asked for the magnitude of the ratio of 𝑞 two to 𝑞 one, we can remove the minus sign we see here. And we see that the magnitude of charge 𝑞 two equals nine times the magnitude of charge 𝑞 one. That’s the charge magnitude ratio that creates this electric field null point.