A long metal cylinder with radius a is supported on an...

A long metal cylinder with radius $a$ is supported on an insulating stand on the axis of a long, hollow, metal tube with radius $b$. The positive charge per unit length on the inner cylinder is $\lambda,$ and there is an equal negative charge per unit length on the outer cylinder. (a) Calculate the potential $V(r)$ for (i) $r<a$; (ii) $a<r<b$; (iii) $r>b$. (Hint: The net potential is the sum of the potentials due to the individual conductors.) Take $V=0$ at $r=b$. (b) Show that the potential of the inner cylinder with respect to the outer is $$ V_{a b}=\frac{\lambda}{2 \pi \epsilon_{0}} \ln \frac{b}{a} $$ (c) Use Eq. (23.23) and the result from part (a) to show that the electric field at any point between the cylinders has magnitude $$ E(r)=\frac{V_{a b}}{\ln (b / a)} \frac{1}{r} $$ (d) What is the potential difference between the two cylinders if the outer cylinder has no net charge?

A long metal cylinder with radius $a$ is supported on an insulating stand on the axis of a long, hollow, metal tube with radius $b$. The positive charge per unit length on the inner cylinder is $\lambda,$ and there is an equal negative charge per unit length on the outer cylinder. (a) Calculate the potential $V(r)$ for (i) $r<a$;
(ii) $a<r<b$; (iii) $r>b$. (Hint: The net potential is the sum of the potentials due to the individual conductors.) Take $V=0$ at $r=b$. (b) Show that the potential of the inner cylinder with respect to the outer is
$$
V_{a b}=\frac{\lambda}{2 \pi \epsilon_{0}} \ln \frac{b}{a}
$$
(c) Use Eq. (23.23) and the result from part (a) to show that the electric field at any point between the cylinders has magnitude
$$
E(r)=\frac{V_{a b}}{\ln (b / a)} \frac{1}{r}
$$
(d) What is the potential difference between the two cylinders if the outer cylinder has no net charge?

Key Concepts

Electric Potential

Electric potential is a scalar quantity that represents the work per unit charge needed to bring a test charge from a reference point (often infinity or a specified point) to a given point in an electric field. It is especially useful in electrostatics where the net potential due to multiple charge distributions can be found by algebraic summation (superposition), and it provides an alternative to directly calculating vector electric fields.

Gauss's Law in Cylindrical Symmetry

Gauss’s law relates the electric flux through a closed surface to the charge enclosed by that surface. In problems with cylindrical symmetry, like the case of coaxial cylinders, Gauss’s law simplifies the calculation of the electric field by choosing a Gaussian surface that mirrors the symmetry of the charge distribution. Once the electric field is known, the potential difference between points can be determined by integrating the field along a suitable path.

Superposition Principle

The superposition principle states that the net electric potential due to several charge distributions is the algebraic sum of the potentials produced by each individual charge distribution. This principle is crucial when dealing with systems where different parts of the configuration (such as an inner charged cylinder and an outer one with opposite charge) contribute independently to the overall potential at any point.

Relationship Between Electric Field and Potential

The electric field is related to the spatial gradient (rate of change) of the electric potential. In regions where the potential varies with position, the electric field is given by the negative derivative of the potential. This relationship is particularly useful when the potential is known as a function of position, allowing one to derive the magnitude and direction of the electric field by differentiation.

Logarithmic Dependence in Cylindrical Systems

In many cylindrical systems, such as coaxial cylinders, the potential and electric field often exhibit a logarithmic dependence on the radial coordinate. This arises from the integration of the 1/r dependence of the field (obtained via Gauss’s law) with respect to the radial distance. The resulting expressions, involving natural logarithms, capture how the potentials change between different cylindrical surfaces.

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