Conductor with charge inside a cavity

imagine we have a spherical conductor with a cavity inside of it and in that cavity you have kept a positive charge let's say it's imagine it's floating in in air over there okay it's and it's not at the center right now the question we want to try and answer is this see this positive charge is going to start producing an electric field and that electric field is going to start sucking the electrons from this metal remember metals have free electrons they can move under the influence of electric field so as a result of that electrons will get sucked in and so there will be some negative charge getting induced over here as a result of that there will be some positive charge getting induced on the outer surface and as a result of that our electric field is going to get modified so the question is once all the electrons have stopped moving once we have reached electrostatic condition we have to answer three questions what would be the total charge on the inner surface and how the distribution would be would be the charge on the outer surface and how the distribution would be and what would the new electric field lines roughly uh the sketch of that how would that look like so these are the three questions we're gonna try and answer so how do we do this where do we even begin this looks like a very complicated challenging puzzle well what i like to do let me first dim the electric field lines a little bit all right so what i like to do is go back to one of the basic things we've seen about conductors at electrostatics you've seen that in electrostatic conduction uh conditions conductors should always have zero electric field inside of them why because if there are any electric fields over here they will start pushing on the electrons and the electrons will start moving that means we are not in electrostatic condition because electrons are moving so whenever we are in electrostatic condition automatically it means that there are no electrons moving automatically it means that the electric field must be zero and if you need a more refresher on you know on on this concept we've talked about this in great detail in a previous video on electrostatic shielding so feel free to go back and check that out but anyways let's come back to our question we know because of this electric field generated by the positive charge it's gonna start you know sucking in negative charges so there will be some negative charges over here and because our conductor is neutral if there are some negative charges which are induced over here because of that some equal number of positive charges should get induced on the outer surface so there should be some positive charges induced on the outer surface and our goal is to figure out how much these are okay because we are given that this inner charge is plus q so let's start with the um induced inner charges the purple charges negative charges how much would that be would it be equal to negative q would it be more than negative q would be less than negative q or do you think it's possible do you think it you know there is no inform not enough information will not be able to calculate that how do we do this how do we figure out how much this negative charges could be well at first it looks like how do i know like how would i know in the electrostatic situation exactly how much of the negative charges would come maybe it depends upon the position of this charge maybe it depends on the strength of this charge i don't know for sure so how can i calculate that well we can use this to our advantage we know that the electric field has to be zero so the negative charges induced must be in such a way that they should kill the electric field over here but how how do i figure out exactly how much what is the connection between charges and electric field gauss's law gauss's law is the one that connects charges in electric field so why don't you pause the video and see if you can somehow use gauss's law to figure out exactly how much negative charge should be induced inside so why don't you pause the video and think about that so what i would like to do is i'm going to choose a gaussian surface and make sure that gaussian surface the entire surface choose a sphere but the entire surface lies within the bulk of the conductor so imagine this is my spherical gaussian surface now because i know that under electrostatic conditions electric field everywhere over here is zero therefore i know that the total flux through that surface must be zero so net electric flux must be zero which means that the total enclosed charge must be zero and from that i think i now understand how much this negative charge should be if the total enclosed charge should be zero and i already have plus q that means that the total negative charge must be negative q so this should be negative q only then the total enclosed charge becomes zero and what's amazing is that it doesn't matter where i position this this little q it doesn't matter where i move it the total induce must be negative q and therefore the total induced on the outer surface should be plus q it has to be because we know that the conductor is neutral so if there is negative q inner induced on the inner surface plus q must in get induced on the outer surface so the charge on the outer surface is plus q the charge on the inner surface is negative q and so what's interesting is whatever charge we introduce inside automatically that charge gets manifested on the outside wonderful isn't it there's an amazing thing about a conductor but we're not done yet we also need to look at the charge distribution and what the electric field looks like what the new electric field is going to look like so let me bring back the original electric field um here it is so this is this would have been the electric field at the beginning before all of these charges got induced the question is what would this new electric field look like let's do this in steps let's first look at the electric field in this region what's gonna happen to that we know that electric field has to be zero in understanding electrostatic conditions so we can go ahead and drop that so let me do this this electric field is going to disappear all right so we've got one part solved the next question i want to answer is how would the new electric field inside look like would it look exactly this way which would just look like this or do you think it's going to change again can you pause the video and think a little bit about this all right we know that because the entire conductor has zero electric field it's an equipotential surface again something that we have talked about in previous videos but because our conductor is an equipotential surface that means the electric field over here must always be perpendicular to the surface and you can see that these electric field lines are not over here they're not perpendicular which means these electric field lines are going to bend in such a way that they try to be perpendicular wherever they meet the surface so the way it's going to look like again it's a rough so rough figure the electric field lines are going to bend somewhat like this from positive to negative but they also always ensure that the angle between the field lines and the inner surface is going to be 90 degrees all right i have another question for you how do you think would the charges get reduced uh this how would the charges charge distribution look like do you think it's gonna be a uniform charge distribution over here or do you think that's to be you know some some part is going to have more concentration while some part is going to have less concentration again can you pause the video and think about how the charge distribution would be on the inner surface this negative q all right so because the charge is closer to this surface and you can see the electric field lines over here are stronger and uh they're close to each other and the electric field over here is far away and theref which means it's weaker we can immediately guess that there should be more charges over here and there should be less so you would have stronger charge distribution i know it's a little hard to see but there should be more charges on this side and there will be less charges on this side so i'll just look at more concentrated over here so this will be less concentrated over here all right final question how would the outer charge distribution look like would it be uniform because it's a sphere or do you think it's going to be um there'll be more positive charges over here because there is more negative charge over here and there will be less positive charge over here because there is less negative charge over here so final time can you pause and think a little bit about how would that be all right now when i first thought about this i thought yeah because there's a lot of negative on this side there would be a lot of positive over here a lot of positive charge over here and there should be less positive charge over here right but think about this there is no electric field in between the outer charges are sort of disconnected from what's going on inside and that's the key which means that the outer charge distribution is absolutely independent of how the inner charge distribution is happening the two are literally disconnected therefore the outer charge distribution does not depend upon the inner charges it only depends upon the geometry and because it's a sphere we've already seen before because of the symmetry the outer charge distribution has to be uniform and as a result this electric field is also going to change slightly remember this electric field now appears to be coming from this charge but because we have a symmetric uniform charge distribution the electric field is going to be radial so it's going to appear to be coming from the center and so the new electric field outer electric field is going to be radial what i absolutely love about this problem is it's really challenging but to answer that we just needed the the core of this is going back to the basics that in electrostatic condition electric field inside a conductor must be zero