An infinitely long cylinder of radius R has linear charge density λ. The potential on the surface of the cylinder is V0, and the electric field outside the cylinder is Er = λ/2πϵ0r . Find the potential relative to the surface at a point that is distance r from the axis, assuming r\u003eR.

Ch 26: Potential and Field

Knight Calc - Physics for Scientists and Engineers 5th Edition

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Knight Calc 5th Edition

Ch 26: Potential and Field

Problema 40

Capitolo 26, Problema 40

An infinitely long cylinder of radius R has linear charge density λ. The potential on the surface of the cylinder is V₀, and the electric field outside the cylinder is E_r = λ/2πϵ₀r . Find the potential relative to the surface at a point that is distance r from the axis, assuming r>R.

Guida verificata passo dopo passo

Start by recalling the relationship between the electric field and the potential in electrostatics. The electric field is the negative gradient of the potential: \( E_r = -\frac{dV}{dr} \). Rearrange this to express the potential difference: \( dV = -E_r \, dr \).

Substitute the given expression for the electric field outside the cylinder, \( E_r = \frac{\lambda}{2\pi\epsilon_0 r} \), into the equation for \( dV \): \( dV = -\frac{\lambda}{2\pi\epsilon_0 r} \, dr \).

Integrate both sides to find the potential \( V(r) \) at a distance \( r \) from the axis. The integral becomes \( V(r) - V_0 = -\int_R^r \frac{\lambda}{2\pi\epsilon_0 r} \, dr \), where the limits of integration are from \( R \) (the surface of the cylinder) to \( r \) (the point of interest).

Evaluate the integral \( \int \frac{1}{r} \, dr \), which is \( \ln(r) \). Substituting this result, the equation becomes \( V(r) - V_0 = -\frac{\lambda}{2\pi\epsilon_0} [\ln(r) - \ln(R)] \).

Simplify the expression using the logarithmic property \( \ln(a) - \ln(b) = \ln(\frac{a}{b}) \). The final expression for the potential relative to the surface is \( V(r) = V_0 - \frac{\lambda}{2\pi\epsilon_0} \ln\left(\frac{r}{R}\right) \).

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Electric Potential

Electric potential is the amount of electric potential energy per unit charge at a point in an electric field. It is a scalar quantity that indicates the work done to move a unit positive charge from a reference point to a specific point in the field. In this context, the potential on the surface of the cylinder is given as V0, which serves as a reference for calculating the potential at a distance r from the axis.

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Electric Field of a Charged Cylinder

The electric field outside an infinitely long charged cylinder is determined by its linear charge density (λ) and is given by the formula Er = λ/(2πϵ0 r). This field decreases with distance from the cylinder and is directed radially outward. Understanding this relationship is crucial for calculating the potential at a distance r from the cylinder's axis.

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In electrostatics, the potential difference between two points can be found by integrating the electric field along a path between those points. For a charged cylinder, this involves integrating the electric field from the surface (r = R) to the point at distance r. This process allows us to determine how the potential changes with distance in the presence of an electric field.

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An infinitely long cylinder of radius R has linear charge density λ. The potential on the surface of the cylinder is V0, and the electric field outside the cylinder is Er = λ/2πϵ0r . Find the potential relative to the surface at a point that is distance r from the axis, assuming r>R.

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Concetti chiave

Electric Potential

Electric Field of a Charged Cylinder

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An infinitely long cylinder of radius R has linear charge density λ. The potential on the surface of the cylinder is V₀, and the electric field outside the cylinder is E_r = λ/2πϵ₀r . Find the potential relative to the surface at a point that is distance r from the axis, assuming r>R.