Question

A solid sphere of radius 40.0 cm has a total positive charge of 17.4 µC uniformly distributed throughout its volume. Calculate the magnitude of the electric field at the following distances. (a) 0 cm from the center of the sphere
..................... kN/C

(b) 10.0 cm from the center of the sphere
..........................kN/C

(c) 40.0 cm from the center of the sphere
........................ kN/C (d) 71.0 cm from the center of the sphere
..................... kN/C

Answer 1

Answer:

(a)Magnitude of the electric field at  0 cm from the center of the sphere:

E=0              

(b) Magnitude of the electric field at 10.0 cm from the center of the sphere :

  E= 244.7 KN/C

(c) Magnitude of the electric field at 40.0 cm from the center of the sphere :

  E=978.8 KN/C

(d) Magnitude of the electric field at 71 cm from the center of the sphere :  

   E=310.7 KN/C

Explanation:

If we have a uniform charge sphere we can use the following formulas to calculate the Electric field due to the charge of the sphere:

$E=\frac{k*Q}{r^{2} }$  : Formula (1) To calculate the electric field in the region outside the sphere r ≥ a

$E: k*\frac{Q}{a^{3} } *r$  :Formula (2) To calculate the electric field in the inner region of the sphere. r ≤ a

Where:

K: coulomb constant

a: sphere radius

Q:  Total sphere charge

r : Distance from the center of the sphere to the region where the electric field is calculated

Equivalences

1μC=10⁻⁶C

1cm= 10⁻²m

Data

k= 9*10⁹ N*m²/C²

Q=17.4 μC=17.4 *10⁻⁶C

a= 40 cm = 40*10⁻²m = 0.4m

Problem development

(a)Magnitude of the electric field at  0 cm :

We replace r=0 in the formula (2) , then, E=0

(b) Magnitude of the electric field at 10.0 cm from the center of the sphere

r<a , We apply the Formula (2):

$E=9*10^{9} *\frac{ 17.4*10^{-6} }{0.4^{3} } *0.1$

E= 244.7 KN/C

(c) Magnitude of the electric field at 40.0 cm from the center of the sphere

r=a, We apply the Formula (1) :

$E=\frac{9*10^{9}*17.4*10^{-6} }{0.4^{2} }$

E=978.8 KN/C

(d) Magnitude of the electric field at 71 cm from the center of the sphere  

r>a , We apply the Formula (1) :

$E= \frac{9*10^{9}*17.4*10^{-6} }{0.71^{2} }$

E=310.7 KN/C