- #1

- 92

- 23

- Homework Statement
- A sphere of radius R carries a total positive charge Q distributed

uniformly throughout its volume. Find the electrostatic

potential inside the sphere.

- Relevant Equations
- Vf - Vi = - integral (E dot dl)

I used the potential at the surface of the sphere for my reference point for computing the potential at a point r < R in the sphere. The potential at the surface of the sphere is ## V(R) = k \frac {Q} {R} ##.

To find the potential inside the sphere, I used the Electric field inside of an insulating sphere from Gauss' Law, ## \vec E = k \frac {Qr} {R^3} \hat r ##.

The potential difference is given by ## \Delta V = -\int_{P_1}^{P_2} \vec E \cdot d \vec l ##

Using this I got $$ V(r) - V(R) = -\int_{R}^{r} \vec E \cdot d \vec l $$

$$ V(r) - V(R) = -\int_{R}^{r} k \frac {Qr} {R^3} \hat r \cdot d \vec l $$ where ## d \vec l ## points from R (outer radius) to r (inner) and is shown in the drawing below.

(I think this might be where I have a problem but I'm not sure why)

Because ##\hat r## and ##d \vec l ## point in opposite directions (they are anti parallel), I have the dot product $$ \hat r \cdot d \vec l = | \hat r || d \vec l | \cos(\pi) = -dr $$ where dr is the magnitude of ## d \vec l ##.

This lead me to $$ V(r) - V(R) = -\int_{R}^{r} k \frac {Qr} {R^3} -dr $$

$$ V(r) - V(R) = \int_{R}^{r} k \frac {Qr} {R^3} dr $$

$$ V(r) - V(R) = k \frac {Q} {R^3} \int_{R}^{r} r dr $$

$$ V(r) - V(R) = k \frac {Q} {R^3} \frac {r^2} {2} - \frac {R^2} {2} $$

$$ V(r) = k \frac {Q} {R^3} * (\frac {r^2} {2} - \frac {R^2} {2}) + V(R) $$

$$ V(r) = k \frac {Q} {R^3} * (\frac {r^2} {2} - \frac {R^2} {2}) + V(R) $$

$$ V(r) = k \frac {Qr^2} {2R^3} - k \frac {Q} {2R} + k \frac {Q} {R} $$

$$ V(r) = k \frac {Qr^2} {2R^3} + k \frac {Q} {2R} $$

This is wrong; my answer implies that the potential would increase as you move away from the positively charged sphere. Also my book says the correct answer is ## V(r) = -k \frac {Qr^2} {2R^3} + k \frac {3Q} {2R} ##. I'm not exactly sure how/what went wrong, but I think it has to do with the dot product ## \vec E \cdot d \vec l ##. Any help would be much appreciated! Thanks in advance!

Diagram showing the sphere of radius R with ## \vec E ## radially outwards. Also indicated ## d \vec l ## pointing from R to r and the positive ## \hat r ## direction pointing radially outwards.

To find the potential inside the sphere, I used the Electric field inside of an insulating sphere from Gauss' Law, ## \vec E = k \frac {Qr} {R^3} \hat r ##.

The potential difference is given by ## \Delta V = -\int_{P_1}^{P_2} \vec E \cdot d \vec l ##

Using this I got $$ V(r) - V(R) = -\int_{R}^{r} \vec E \cdot d \vec l $$

$$ V(r) - V(R) = -\int_{R}^{r} k \frac {Qr} {R^3} \hat r \cdot d \vec l $$ where ## d \vec l ## points from R (outer radius) to r (inner) and is shown in the drawing below.

(I think this might be where I have a problem but I'm not sure why)

Because ##\hat r## and ##d \vec l ## point in opposite directions (they are anti parallel), I have the dot product $$ \hat r \cdot d \vec l = | \hat r || d \vec l | \cos(\pi) = -dr $$ where dr is the magnitude of ## d \vec l ##.

This lead me to $$ V(r) - V(R) = -\int_{R}^{r} k \frac {Qr} {R^3} -dr $$

$$ V(r) - V(R) = \int_{R}^{r} k \frac {Qr} {R^3} dr $$

$$ V(r) - V(R) = k \frac {Q} {R^3} \int_{R}^{r} r dr $$

$$ V(r) - V(R) = k \frac {Q} {R^3} \frac {r^2} {2} - \frac {R^2} {2} $$

$$ V(r) = k \frac {Q} {R^3} * (\frac {r^2} {2} - \frac {R^2} {2}) + V(R) $$

$$ V(r) = k \frac {Q} {R^3} * (\frac {r^2} {2} - \frac {R^2} {2}) + V(R) $$

$$ V(r) = k \frac {Qr^2} {2R^3} - k \frac {Q} {2R} + k \frac {Q} {R} $$

$$ V(r) = k \frac {Qr^2} {2R^3} + k \frac {Q} {2R} $$

This is wrong; my answer implies that the potential would increase as you move away from the positively charged sphere. Also my book says the correct answer is ## V(r) = -k \frac {Qr^2} {2R^3} + k \frac {3Q} {2R} ##. I'm not exactly sure how/what went wrong, but I think it has to do with the dot product ## \vec E \cdot d \vec l ##. Any help would be much appreciated! Thanks in advance!

Diagram showing the sphere of radius R with ## \vec E ## radially outwards. Also indicated ## d \vec l ## pointing from R to r and the positive ## \hat r ## direction pointing radially outwards.