Verifying the Electric Field due to a Shell of Charge [20pts]. We can show that, for a uniformly charged, very thin spherical shell centered at the origin, the electric field is given by: 10 rR where Q is the total charge on the sphere, and R is the radius of the sphere. (Note that the electric field is undefined for r= R.) In other words, inside the sphere the electric field is zero, and outside the sphere the electric field is the same as that for a point charge Q located at the origin. We would like to prove this relationship numerically using MATLAB. Design Specifications 1. Find or derive an expression for a differential element of charge dq in spherical coordinates. 2. Derive an expression for the rectangular components of the differential vector field dE at an arbitrary point in space r = (x, y, z) due to the differential element dq, located at a particular coordinate (R, 0, 0) on the surface of the sphere. That is, the expression should depend on these six quantities (plus k and Q), and should be in rectangular form, dE = (dEx, dEy, dEz), as this is the only way to add up the contributions from every differential element of charge dq. 3. Write a MATLAB script to numerically compute the total electric field at an arbitrary point in space, either inside or outside the spherical shell. Use R = 2m and Q = 5C. The coordinates (x, y, z) should be user inputs. The range of should have around 1000 points, and the range of should have twice as many points as the range of e. (And be careful with "points" vs. "intervals".) You may use k = 1 throughout, as this will make the output much easier to interpret. You should have two nested for loops for the numerical calculation of the total electric field. 4. Check your hand calculation first by computing the total charge on the sphere. In other words, if the numerical sum of every dq is not equal to the total charge , then there is no reason to compute the electric field. This will make it more likely that your differential charge element da is correct and that your nested for loops are set up correctly. 5. Check your calculation also using the given function E(r) as the analytic solution. In general, a proper check returns zero. (And in this case, a proper check returns an array of three zeros)