Answer :
To find the value of the charge given the electric potential and the distance, we use the formula for electric potential due to a point charge:
[tex]\[ V = \frac{k \cdot Q}{r} \][/tex]
where:
- [tex]\( V \)[/tex] is the electric potential (580 V),
- [tex]\( k \)[/tex] is Coulomb's constant, [tex]\( k = \frac{1}{4 \pi \varepsilon_0} \)[/tex], with [tex]\( \varepsilon_0 \)[/tex] being the permittivity of free space ([tex]\( \varepsilon_0 = 8.854 \times 10^{-12} \, \text{F/m} \)[/tex]),
- [tex]\( Q \)[/tex] is the charge,
- [tex]\( r \)[/tex] is the distance from the charge (1.34 m).
First, solve for [tex]\( Q \)[/tex]:
[tex]\[ Q = V \cdot r \cdot 4 \pi \varepsilon_0 \][/tex]
Given the values:
[tex]\[ V = 580 \, \text{V} \][/tex]
[tex]\[ r = 1.34 \, \text{m} \][/tex]
[tex]\[ \varepsilon_0 = 8.854 \times 10^{-12} \, \text{F/m} \][/tex]
[tex]\[ Q = 580 \cdot 1.34 \cdot 4 \pi \cdot 8.854 \times 10^{-12} \][/tex]
Calculating the product of the constants, we get approximately:
[tex]\[ Q \approx 8.647332802006347 \times 10^{-8} \, \text{C} \][/tex]
So, the value of the charge is:
[tex]\[ \boxed{8.647332802006347} \times 10^{-8} \, \text{C} \][/tex]
Since the problem asks only for the number, the value is:
[tex]\[ \boxed{8.647332802006347} \][/tex]
However, for clarity and considering the simplified presentation for practical purposes, this is often rounded to:
[tex]\[ \boxed{8.647} \, \times 10^{-8} \, \text{C} \][/tex]
[tex]\[ V = \frac{k \cdot Q}{r} \][/tex]
where:
- [tex]\( V \)[/tex] is the electric potential (580 V),
- [tex]\( k \)[/tex] is Coulomb's constant, [tex]\( k = \frac{1}{4 \pi \varepsilon_0} \)[/tex], with [tex]\( \varepsilon_0 \)[/tex] being the permittivity of free space ([tex]\( \varepsilon_0 = 8.854 \times 10^{-12} \, \text{F/m} \)[/tex]),
- [tex]\( Q \)[/tex] is the charge,
- [tex]\( r \)[/tex] is the distance from the charge (1.34 m).
First, solve for [tex]\( Q \)[/tex]:
[tex]\[ Q = V \cdot r \cdot 4 \pi \varepsilon_0 \][/tex]
Given the values:
[tex]\[ V = 580 \, \text{V} \][/tex]
[tex]\[ r = 1.34 \, \text{m} \][/tex]
[tex]\[ \varepsilon_0 = 8.854 \times 10^{-12} \, \text{F/m} \][/tex]
[tex]\[ Q = 580 \cdot 1.34 \cdot 4 \pi \cdot 8.854 \times 10^{-12} \][/tex]
Calculating the product of the constants, we get approximately:
[tex]\[ Q \approx 8.647332802006347 \times 10^{-8} \, \text{C} \][/tex]
So, the value of the charge is:
[tex]\[ \boxed{8.647332802006347} \times 10^{-8} \, \text{C} \][/tex]
Since the problem asks only for the number, the value is:
[tex]\[ \boxed{8.647332802006347} \][/tex]
However, for clarity and considering the simplified presentation for practical purposes, this is often rounded to:
[tex]\[ \boxed{8.647} \, \times 10^{-8} \, \text{C} \][/tex]
