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To find the charge that produces an electric field strength of 40 V/cm at a distance of 30 cm in a vacuum, we can use Coulomb’s law. The electric field ((E)) at a distance (r) from a point charge (q) in a vacuum is given by the formula:
[E = frac{kq}{r^2}]
where (k) is Coulomb’s constant ((8.987 times 10^9 , text{N m}^2/text{C}^2)), (q) is the charge in coulombs (C), and (r) is the distance from the charge in meters.
Given that the electric field strength (E = 40, text{V/cm} = 4000, text{V/m}) (since 1 V/m = 0.01 V/cm), and the distance (r = 30, text{cm} = 0.3, text{m}), we can rearrange the formula to solve for (q):
[q = frac{E cdot r^2}{k}]
Substitute the given values into the equation:
[q = frac{4000 cdot (0.3)^2}{8.987 times 10^9}]
[q = frac{4000 cdot 0.09}{8.987