Student handout: Total Charge

Static Fields 2024 (9 years)
In this small group activity, students integrate over non-uniform charge densities in cylindrical and spherical coordinates to calculate total charge.

Calculating Total Charge

Each group will be given one of the charge distributions given below: ($\alpha$ and $k$ are constants with dimensions appropriate for the specific example.)

• Spherical Symmetery
1. A positively charged (dielectric) spherical shell of inner radius $a$ and outer radius $b$ with a spherically symmetric internal charge density $\rho (\vec{r}) = \alpha\, r^{3}$
2. A positively charged (dielectric) spherical shell of inner radius $a$ and outer radius $b$ with a spherically symmetric internal charge density $\rho (\vec{r}) =\alpha\, e^{(kr)^{3}}$
3. A positively charged (dielectric) spherical shell of inner radius $a$ and outer radius $b$ with a spherically symmetric internal charge density $\rho (\vec{r}) = \alpha\, \frac{1}{r^{2}}\, e^{(kr)}$
• Cylindrical Symmetry
1. A positively charged (dielectric) cylindrical shell of inner radius $a$ and outer radius $b$ with a cylindrically symmetric internal charge density $\rho (\vec{r}) = \alpha\, s^{3}$
2. A positively charged (dielectric) cylindrical shell of inner radius $a$ and outer radius $b$ with a cylindrically symmetric internal charge density $\rho (\vec{r}) =\alpha\, e^{(ks)^{2}}$
3. A positively charged (dielectric) cylindrical shell of inner radius $a$ and outer radius $b$ with a cylindrically symmetric internal charge density $\rho (\vec{r}) = \alpha\, \frac{1}{s}\, e^{(ks)}$

1. Find the total charge. (If the total charge is infinite, decide what you should calculate instead to provide a meaningful answer.)
2. Find the dimensions of the constants $\alpha$ and $k$.

Author Information
Corinne Manogue, Tevian Dray
Keywords

Learning Outcomes