A nonconducting sphere of radius 10 cm is charged uniformly with a density of 100 nC/m3. What is the magnitude of the potential difference b

A nonconducting sphere of radius 10 cm is charged uniformly with a density of 100 nC/m3. What is the magnitude of the potential difference between the center and a point 4.0 cm away?

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  1. Explanation:

    The given data is as follows.

       Radius of the sphere (R) = 10 cm = 0.01 m   (as 1 m = 100 cm)

       Distance from the center (r) = 4 cm = 0.04 m

       Charge density ([tex]\sigma[/tex]) = 100 [tex]nC/m^{3}[/tex]

                                      = [tex]100 \times 10^{-9} C/m^{3}[/tex]   (As 1 nm = [tex]10^{-9} m[/tex])

    As the relation between charge and potential difference is as follows.

              Q = [tex]\sigma V[/tex]

                  = [tex]\sigma (\frac{4}{3} \pi r^{3})[/tex]

                  = [tex]100 \times 10^{-9} C/m^{3} \times \frac{4}{3} \pi (0.01)^{3}[/tex]  

                  = [tex]4.19 \times 10^{-10} C[/tex]

    Expression for electric field is as follows.

               E(r) = [tex]k \frac{qr}{R^{3}}[/tex]

    Electric potential, V(r) = [tex]-\int_{0}^{r} E(r) dr[/tex]

                   = [tex]-\int_{0}^{r} k \frac{qr}{R^{3}} dr[/tex]

                   = [tex]-k (\frac{qr^{2}}{2R^{3}})[/tex]  

                   = [tex]-9 \times 10^{9} Nm^{2}/C^{2} (\frac{4.19 \times 10^{-10} C (0.04)^{2}}{2(0.1)^{3}})[/tex]  

                  = -3 V

    Thus, we can conclude that the magnitude of the potential difference between the center and a point 4.0 cm away is -3 V.

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