WEBVTT
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in this question. We have a solid insulating sphere
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off radios a and Villa Metric charge Density Rho,
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whose center is not located at the origin off the
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reference frame and have to calculate what is the electric
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field inside the sphere. For that, we begin
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by defining a new reference frame whose origin coincides with
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the center off that solves fear. So let me
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do it as follows we can define and rather y
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axis that I'm calling. Why prime such that it
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crosses the X axis precisely at the center off this
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fear like this. So this is my Y prime
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access. You can see that the origin off the
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reference frame defined by white, prime and ex coincides
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with the center off that salt sphere. Now we
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can easily calculate using gases Law Wadi ist the electric
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field inside that sphere. Gazans long tells us that
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the flux off electric field is given by the integral
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off D eight scaler Eat where D is a surface
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area element and E is the electric field. I
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noticed that there's a symmetry in this question because we
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have a sphere and this fear we produce electric field
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that points radio. The So It points like this
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always in radial directions. At the same time,
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you can see that the surface off that sphere has
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a normal vector. That is out, sir,
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at these directions. So D is a radio factor
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, and Suri's the electric field. It means that
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they are parole, so the eight is parallel to
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eat. And these implies that the eighth Scaler in
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is the Times E because they are part of vectors
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. Then you can remember that the magnitude off the
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electric field do not depend on the area off that
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Gulshan surface. Then it is egos through the magnitude
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of terrific field times, the integral off D over
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the area off the ocean surface. But the integral
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off D A. Over the area is just the
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surface area. Therefore, the foots off electric field
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is given by the magnitude off the electric field times
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the area off big ocean surface. Then the magnitude
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of the electric field is given by the flux,
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divided by the area off big ocean surface. But
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remember that the flux off the electric field through the
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surface off the Gulshan surface is also equals two.
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Then close a charge Q divided by absolute zero.
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So the magnitude off the electric field is given by
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the enclosed charge divided by the area of the ocean
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times. Absolute zero. But them what is then
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cause a charge? For that we have to peek
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a point. Suppose that we are working, for
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instance, In this point, we want to know
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what is the magnitude of the electric field. At
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this point, let me say that this point that
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a distance r from the origin so disease are Then
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we can pick big ocean surface that is, a
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sphere off radials are like this on. Then we
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can use the equation for the area office. Fear
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the area office Fear is given by four pi r
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squared then the electric field here has a magnitude given
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by Q. Divided by 45 R squared times.
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Absolute zero. Now we have to calculate what is
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cute. Q. Is the quantity of charge that
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is located inside off this guy ocean surface. Notice
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that we have a fella metric charge density Rho,
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which is uniforms the reform. We can use the
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fact that the charge is given by the charge density
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times. The volume in this case, we have
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to use the volume off the ocean surface, which
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is the volume office. Fear the value office Fear
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is given by four pi r two the third divided
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by treat Then then close it charge is rule times
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four pi aren't in the toilet divided by three,
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then a magnitude off the electric field Inside That fear
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is given by rope Times four by are to the
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third divided by three times four times Spy times are
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to the second absolute zero. Now we simplify whatever
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you can't So for planning for pie And here we
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have our tittered so we can simply fight We farmed
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in the second. Then we're last twist An electric
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field with a magnitude given by rope Times are divided
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by three times absolute zero This is the magnitude off
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the electric field in our reference frame. Why prime
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X? Because of the symmetry of this problem,
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it is easy to write this You know that perform
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. We know that the electric field will be parallel
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to the surface area of actor. So the electric
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field vector is given by three absolute zero times are
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so it is proportional to the event for our which
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points in the radial direction. Now, how can
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you translate these, Rector two of actor off.
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Why X Have to notice the following The only difference
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between why prime X and y axis is this vector
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. Let me call it the factor. Be so
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in order to bring these vectors to find it in
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white Prime X back to the origin off. Why
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X All we have to do Sue attract be from
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them. For instance, Here we have our So
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in order to bring this vector are back to the
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y X reference frame. We have to subtract b
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from it. So we end up with an electric
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field that is given by road divided by three times
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at zero times are minus beat the miners be term
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is what brings the vector back toe So original reference
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frame Now that Norland as the board so we can
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proceed to the next item in the next item we
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have these atop we have a big sphere off Radios
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are on uniformed volumetric charged into the room but it
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has a hole inside and it's whole is also a
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sphere off radios eight. That is not located at
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center off the bigs. From here, it's added
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the sense be from the center off that sphere.
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Then we have to complete what is the electric field
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in these region in the region off the whole.
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To solve this question, we can use the superposition
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principle off electrodynamics to treat the problem as follows.
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So instead of treating as fear withhold, we treat
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two spheres. One solid sphere off radios are in
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uniform. Votomatic charge density route that has no hole
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. Plus, as most fear, that is equivalent
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to the whole. So it's not located at the
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origin, but at a distance be from the origin
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. It has a radios A and very important.
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It has a volumetric charge. Density there is uniform
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and given by miners route so that when we super
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pose, this fear will be this year. What
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we got is the situation where we have a region
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inside the big sphere that has a net charge equals
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20 and then the situation is very simple to treat
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. We already know what is the electric field.
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In the first situation. It's given by E one
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, and e one is echoes to roll, divided
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by three times absolute zero times are. Why do
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I know it? Why is it like this?
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I'm using this equation, but in this case the
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center off the sphere coincide with the origin off.
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The reference frame survey is close to zero and that's
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it. Nothing more. So this is the electric
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field. In the first situation for the second situation
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, the electric field is each you and it's given
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by miners rope divided by three times absolute zero times
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are miners meet. And then, using the superposition
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principle, we concluded that the electric field inside the
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whole E is equals two e one plus each year
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on Deasy's room, divided by three times absolute zero
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times are plus minus rope. Times are mine is
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me divided by three times absolute zero. These gives
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us rope. Times are divided by three times absolute
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zero miners. Roll times are divided by three times
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absolute zero. Plus I'm using the distributive property of
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multiplication Here roll times meat divided by three times absolute
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zero. We can see that there is a cancellation
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happening. So this term councils this turn and we
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end up with an electric field that is even by
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rope divided by three times absolute zero Pancks be so
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this is the electric field inside the whole notice that
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it only depends on B and B's a constant,
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so the electric field inside the hole is constant.