A spherical capacitor contains a charge of 3.30 nC when connected to a potential difference of 220 V. If its plates are separated by vacuum and the inner radius of the outer shell is 4.00 cm, calculate: (a) the capacitance; (b) the radius of the inner sphere; (c) the electric field just outside the surface of the inner sphere.

Video transcript

Welcome back everybody. We are making observations about a spherical capacitor. Now this spherical capacitor consists of a inner brass ball and a concentric brass shell. And we are told a couple different things here. We're told that the radius of the inner ball is just right now unknown. We're just going to represent it with a radius of one. We're told that the radius of the shell however is 5.1 centimeters or 0. m. We are told that the voltage applied across the capacitor is 100 and 80 volts and we are told the charges of each element of this capacitor. The charge of the ball is going to be five times 10 to the negative ninth columns. Given by five nano columns. And the charge of the concentric cell is negative five nano columns or negative five times 10 to the negative ninth columns. And we are tasked with finding three different things here. One, what is the capacitance across this capacitor? To what is the value of our inner radius and three. What is the electric field at the surface of the center ball? Alright, so let's go ahead and start with part A. Here, Part A is simply going to be given by this formula we're going to have that are capacitance is going to be the charge of the center ball divided by the voltage. This is going to be five times 10 to the negative ninth divided by 1 80 giving us a capacity of 2. PICO. Their ads great. Moving on to Part B Here, we actually have a formula for capacitance. For this type of capacitor. It is given by four pi epsilon. Not times radius one times radius two divided by radius two minus radius one. So let's go ahead and plug in all the values here and see if we can't simplify from there. We have that. This is equal to 2.78 times 10 to the negative 11th or our capacitance, which is what we found equal to four high times 8.85 times to the negative 12. All times are one times 10.51 divided by uh let's see here 0.51 minus R one. If I divide both sides of this equation by this term right here we then get the following that 0. R one divided by 10.51 minus R one is equal to zero .25 from here. With a little bit of algebraic manipulation. We get that our inner radius is equal .04-3 m or 4.23 cm. Great. So let's go ahead and move on to part C here. I'm gonna scroll down just a little bit for part C. We need to find the electric field at the surface of this center ball right here. So this is simply going to be according to Coolum slaw. Now why can we apply columns law? If we're looking at the central ball here, we can just think about it as a point charge inside of the spherical shell. Right? So this is just going to be K. Times each charge of our ball divided by the radius of our ball. Let's go ahead and plug in those values. Here we have 8.99 times 10 to the ninth times five times 10 to the negative ninth, all divided by the radius. We just found a 0.423 squared, giving us an electric field of 2.51 times 10 to the fourth. Newtons per. So now we have found the capacitance of our capacitor. We have found the inner radius and we have found the electric field at the surface of the central ball. And this all corresponds to our answer choice of B. Thank you all so much for watching. Hope this video helped. We will see you all in the next one.