A very long insulating cylinder of charge of radius 2.50 cm carries a uniform linear density of 15.0 nC/m. If you put one probe of a voltmeter at the surface, how far from the surface must the other probe be placed so that the voltmeter reads 175 V?

Verified step by step guidance

1. Identify the given quantities: radius of the cylinder (r) = 2.50 cm, linear charge density (\(\lambda\)) = 15.0 nC/m, and the voltage difference (V) = 175 V.
2. Convert all units to SI units: radius r = 0.025 m, linear charge density \(\lambda = 15.0 \times 10^{-9} \, \text{C/m}\).
3. Use Gauss's Law to find the electric field (E) outside the cylinder. For a long charged cylinder, the electric field at a distance x from the surface is given by \(E = \frac{\lambda}{2\pi\epsilon_0 (r + x)}\), where \(\epsilon_0\) is the permittivity of free space.
4. Set up the integral of the electric field to find the voltage difference, V, between two points. The voltage difference between the surface of the cylinder and a point x distance away is given by \(V = \int_{r}^{r+x} E \, dr\).
5. Solve the integral for x, which is the distance from the surface where the voltage difference equals 175 V. Rearrange and solve the equation \(175 = \int_{0.025}^{0.025+x} \frac{15.0 \times 10^{-9}}{2\pi \times 8.85 \times 10^{-12} (0.025 + r)} \, dr\) to find the value of x.

Key Concepts

Electric Field of a Charged Cylinder

The electric field (E) outside a long, uniformly charged cylinder can be calculated using Gauss's law. For a cylinder with linear charge density λ, the electric field at a distance r from the axis (where r is greater than the radius of the cylinder) is given by E = λ / (2πε₀r), where ε₀ is the permittivity of free space. This concept is crucial for understanding how the electric field behaves around charged objects.

Voltage and Electric Potential Difference

Voltage, or electric potential difference (V), between two points in an electric field is defined as the work done per unit charge to move a charge between those points. The relationship between electric field and voltage is given by V = -∫E·dr, where the integral is taken along the path from one point to another. This concept is essential for determining how far the second probe must be placed to achieve a specific voltage reading.

Gauss's Law

Gauss's law relates the electric flux through a closed surface to the charge enclosed by that surface. It states that the total electric flux Φ through a closed surface is equal to the enclosed charge Q divided by the permittivity of free space ε₀: Φ = Q/ε₀. This principle is fundamental in deriving the electric field around symmetric charge distributions, such as the long insulating cylinder in this problem.