Answer :
The radius of the sphere is r=5.15 cm=0.0515 m, and its surface is given by
[tex]A=4 \pi r^2 = 4 \pi (0.0515 m)^2 = 0.033 m^2[/tex]
So the total charge on the surface of the sphere is, using the charge density
[tex]\rho=+1.21 \mu C/m^2 = +1.21 \cdot 10^{-6} C/m^2[/tex]:
[tex]Q= \rho A = (+1.21 \cdot 10^{-6} C/m^2)(0.033 m^2)=4.03 \cdot 10^{-8}C[/tex]
The electrostatic force between the sphere and the point charge is:
[tex]F=k_e \frac{Qq}{r^2} [/tex]
where
ke is the Coulomb's constant
Q is the charge on the sphere
[tex]q=+1.75 \muC = +1.75 \cdot 10^{-6}C[/tex] is the point charge
r is their separation
Re-arranging the equation, we can find the separation between the sphere and the point charge:
[tex]r=\sqrt{ \frac{k_e Q q}{F} }= \sqrt{ \frac{(8.99 \cdot 10^9 Nm^2 C^{-2})(4.03 \cdot 10^{-8} C)(1.75 \cdot 10^{-6}C)}{35.9 \cdot 10^{-3}N} }=0.133 m=13.3 cm [/tex]
[tex]A=4 \pi r^2 = 4 \pi (0.0515 m)^2 = 0.033 m^2[/tex]
So the total charge on the surface of the sphere is, using the charge density
[tex]\rho=+1.21 \mu C/m^2 = +1.21 \cdot 10^{-6} C/m^2[/tex]:
[tex]Q= \rho A = (+1.21 \cdot 10^{-6} C/m^2)(0.033 m^2)=4.03 \cdot 10^{-8}C[/tex]
The electrostatic force between the sphere and the point charge is:
[tex]F=k_e \frac{Qq}{r^2} [/tex]
where
ke is the Coulomb's constant
Q is the charge on the sphere
[tex]q=+1.75 \muC = +1.75 \cdot 10^{-6}C[/tex] is the point charge
r is their separation
Re-arranging the equation, we can find the separation between the sphere and the point charge:
[tex]r=\sqrt{ \frac{k_e Q q}{F} }= \sqrt{ \frac{(8.99 \cdot 10^9 Nm^2 C^{-2})(4.03 \cdot 10^{-8} C)(1.75 \cdot 10^{-6}C)}{35.9 \cdot 10^{-3}N} }=0.133 m=13.3 cm [/tex]
Final answer:
The separation between the point charge and the center of the sphere is approximately [tex]\( 0.0281 \, \text{m} \).[/tex]
Explanation:
Using Coulomb's law, we have:
[tex]\[ F = \frac{{k \cdot |q_1| \cdot |q_2|}}{{r^2}} \][/tex]
Where:
- F is the electrostatic force,
- k is Coulomb's constant [tex](\( 8.99 \times 10^9 \, \text{N m}^2/\text{C}^2 \)),[/tex]
- [tex]\( q_1 \)[/tex]and [tex]\( q_2 \)[/tex] are the magnitudes of the charges, and
- r is the separation between the charges.
Rearranging for r, we get:
[tex]\[ r = \sqrt{\frac{{k \cdot |q_1| \cdot |q_2|}}{{F}}} \][/tex]
Substituting the given values:
[tex]\[ r = \sqrt{\frac{{8.99 \times 10^9 \, \text{N m}^2/\text{C}^2 \cdot (12.1 \times 10^{-6} \, \text{C/m}^2) \cdot (1.75 \times 10^{-6} \, \text{C})}}{{35.9 \times 10^{-3} \, \text{N}}}} \][/tex]
[tex]\[ r = \sqrt{\frac{{8.99 \times 10^9 \cdot 12.1 \cdot 1.75}}{{35.9}}} \, \text{m} \][/tex]
[tex]\[ r \approx 0.0281 \, \text{m} \][/tex]
Therefore, the separation between the point charge and the center of the sphere is approximately[tex]\( 0.0281 \, \text{m} \).[/tex]
