Is the Electric Field Calculation Consistent with the Potential Result?

[Cactus]

Homework Statement

Hey, I just wanted to post these two questions to check whether the answers I've gotten are correct, as I have no real way of checking if what I've done is correct (This is for parts c and d of the attached question)

For c, would the electric field along the z axis just be the superposition of the field from a negatively charged plate plus the field of a positively charged sphere (as along the z axis the field lines are parallel to the axis)

Likewise, for d, would this be the correct way to go about solving for capacitance given the formula for potential, as I can't see any other way to cancel the Vo and express capacitance purely in geometric terms

Relevant Equations

Capacitance = Charge/Potential Difference
Electric Field of an Infinite Plate
Electric Field of a Sphere

Question

Part C

Part D

 

[Masano Hutama]

at part C How you can write $$ E_{-}=\frac{|\pi|}{2\epsilon_0} $$ and what is $$ \pi $$

 

[Cactus]

at part C How you can write $$ E_{-}=\frac{|\pi|}{2\epsilon_0} $$ and what is $$ \pi $$

ah that's not pi its just a badly written n as in surface charge density

 

[Delta2]

Part c must be correct from you, only thing I have to say is that the d in the book scheme and the d in your notes are different. If I call d' the d of your notes and simply d the d from your book it is

d=d'+R, where R is the radius of the solid sphere,

and having that in mind, the result you get for electric field E in part c seems to be in agreement with the result for potential V that is given by the book(Just integrate your E-field and you ll get the V as presented by the book, we know that $V=\int Edr$).

Part d also seems correct.

 

[Cactus]

Part c must be correct from you, only thing I have to say is that the d in the book scheme and the d in your notes are different. If I call d' the d of your notes and simply d the d from your book it is

d=d'+R, where R is the radius of the solid sphere,

and having that in mind, the result you get for electric field E in part c seems to be in agreement with the result for potential V that is given by the book(Just integrate your E-field and you ll get the V as presented by the book, we know that $V=\int Edr$).

Part d also seems correct.

Yeah I realized my mistake with the d and d' after posting this and fixed that, but thanks for the reply and confirmation on answers. I've also had a friend finish this question now and got the same answers so fingers crossed they're right