Hi, can someone provide some suggestions? I'm stuck on the following questions.(adsbygoogle = window.adsbygoogle || []).push({});

Q. In this question we will consider the consequences if photons had mass. For massive photons the Laplace equation for the electric potential is replaced by [itex]\nabla ^2 \Phi = m^2 \Phi [/itex]. (*)

a) Use spherical coordinates to show that the Yukawa potential given by

[tex]\Phi = \frac{{qe^{ - mr} }}{r}[/tex]

is a solution of (*). Here q and m are constants. (Note: The Yukawa potential thus replaces the Coulomb potential for a point charge at the origin.)

b) Find the corresponding Yukawa electric field

[tex]\mathop E\limits^ \to = E\left( r \right)\mathop r\limits^ \to = - \nabla \Phi [/tex]. (The r is supposed to be a unit vector)

c) Is div(E) = 0 in regions excluding the origin, i.e. in regions without charge? Is curl(E) = 0?

d) Calculate:

[tex]\int\limits_C^{} {\mathop E\limits^ \to \bullet d\mathop s\limits^ \to } [/tex] where C is the curve paramterised by

[tex]\left( {r,\theta ,\phi } \right) = \left( {1 + 2\cos t,3t,2t} \right),t \in \left[ {0,2\pi } \right][/tex].

Here is what I did.

In part a, I worked out the Laplacian in spherical coordinates, said something about the independence of [itex]\Phi [/itex] on theta and phi, calculated the derivative of [itex]\Phi [/itex] with respect to r and showed that it equaled the RHS. Is that correct?

Part b: In part 'a' I calculated the derivative of [itex]\Phi [/itex] with respect to r (the derivatives with respect to theta and phi are zero).

[tex]

\frac{{\partial \Phi }}{{\partial r}} = - q\left( {\frac{{mr + 1}}{{r^2 }}} \right)e^{ - mr}

[/tex]

So

[tex]

\mathop E\limits^ \to = - \nabla \Phi = \left( {q\left( {\frac{{mr + 1}}{{r^2 }}} \right)e^{ - mr} } \right)\mathop r\limits^ \to

[/tex] ?

c) I did this by actually calculating the curl and divergence. Is there an easier way? (I found that the divergence is non-zero and the curl is zero)

d) Ok at first glance, two problems.

(1) I don't get what's with the [itex]d\mathop s\limits^ \to [/itex]. Isn't ds normally reserved for arc length? If its a line integral then shouldn't the ds be a dr or a dx?

(2) Ignoring (1) and taking the ds to be dx = c'(t)dt then we have a line integral. Evaluating E(c(t)) and calculating the line integral looks impossible. But since E looks like it's conservative (it's the gradient of a scalar function), I was thinking maybe make use of the function [itex]d\mathop s\limits^ \to [/itex].

To this end, I would calculate c(0) = (3, 0, 0) and c(pi/2) = (2, ..., ...). Can I now plug these values the following?

[itex]\Phi = \frac{{qe^{ - mr} }}{r}[/itex]

It's just that this question involves spherical coordinates so I'm not sure if the [itex]\Phi \left( {final} \right) - \Phi \left( {initial} \right)[/itex] approach is valid.

Any help would be good thanks.

Note: This is not from a physics subject so no knowledge of physics should be required.

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# Homework Help: Vector calculus

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