Determine at what distance from the charged cylinder

What is the distance from a charged cylinder at which the field strength is 410^5 V/m? The diameter of the cylinder is 4 cm, and the surface charge density is 8.8510^-6 C/m^2.

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To solve the problem, it is necessary to use the formula for the electric field strength of a point charge:

E = k * Q / r^2,

where E is the field strength, Q is the charge of the cylinder, r is the distance from the cylinder to the point at which the field strength is measured, k is the electrical constant.

For a cylinder with surface charge density σ, charge Q can be calculated using the formula:

Q = σ * S,

where S is the surface area of ​​the cylinder.

For a cylinder with diameter d, the surface area can be found using the formula:

S = π * d^2 / 4.

Thus, the charge of the cylinder will be equal to:

Q = σ * π * d^2 / 4.

We substitute the values ​​into the formula for field strength:

E = k * σ * π * d^2 / (4 * r^2).

We substitute the known values: E = 4 * 10^5 V/m, d = 4 cm = 0.04 m, σ = 8.85 * 10^-6 C/m^2, k = 1 / (4πε0) = 9 * 10^9 N * m^2 / C^2, where ε0 is the dielectric constant of vacuum.

We get the equation:

4 * 10^5 = 9 * 10^9 * 8.85 * 10^-6 * π * (0.04)^2 / (4 * r^2).

We solve the equation for r:

r = sqrt(9 * 10^9 * 8.85 * 10^-6 * π * (0.04)^2 / (4 * 4 * 10^5)) ≈ 0.015 m.

Answer: the distance from the charged cylinder at which the field strength is 4*10^5 V/m is about 1.5 cm.

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