Capacitors with Calculus: Videos & Practice Problems

Capacitors with Calculus extends the basic idea of capacitance to shapes where the electric field is not uniform. The central relationship is \(C=\frac{Q}{V}\) , but for curved geometries the voltage must be found from \(V=-\int_A^B \vec{E}\cdot d\vec{l}\) . This is needed for conductors such as concentric spheres or cylinders, where the field changes with distance.

For a spherical capacitor made of two concentric conducting shells, the field between shells follows the enclosed charge, so the capacitance depends on the radii of the inner and outer shells. For concentric cylindrical shells, the same calculus approach gives a result in terms of capacitance per unit length, since an infinitely long cylinder would otherwise have infinite total capacitance. In both cases, the key skill is combining electric field, potential difference, and geometry to derive capacitance from the region between the charged surfaces.

Capacitance of Special Capacitor Shapes Using Calculus

Capacitance is fundamentally defined as the ratio of the charge Q on a conductor to the potential difference V between two charged surfaces, expressed as C = Q / V. While calculating capacitance for parallel plate capacitors is straightforward due to their uniform electric field, more complex geometries like spherical or cylindrical capacitors require calculus to account for the varying electric field.

In a parallel plate capacitor, the electric field E is constant between the plates, so the potential difference is simply the product of the electric field and the distance between the plates, V = -E × d. However, for shapes such as concentric spheres, the electric field changes with distance because it depends on the enclosed charge and the geometry of the surfaces. This variation means the electric field is not uniform, and the potential difference must be calculated using the integral form:

\[ V = -\int_{A}^{B} \mathbf{E} \cdot d\mathbf{l} \]

where the integral is taken along the path from one surface to the other, and d𝑙 is the infinitesimal displacement vector. The dot product accounts for the component of the electric field along the path.

Consider the classic example of a spherical capacitor composed of two concentric spherical conducting shells with radii A (inner shell) and B (outer shell), carrying charges +Q and -Q respectively. The electric field between these shells behaves like that of a point charge and is radially directed. Its magnitude at a distance r from the center is given by Coulomb’s law:

\[ E(r) = \frac{kQ}{r^2} \hat{r} \]

where k is Coulomb’s constant, k = \(\frac{1}{4\pi \varepsilon_0}\), and 𝜀₀ is the permittivity of free space.

To find the potential difference between the shells, integrate the electric field from radius A to B:

\[ V_{AB} = -\int_{A}^{B} \mathbf{E} \cdot d\mathbf{l} = -\int_{A}^{B} \frac{kQ}{r^2} dr = kQ \left( \frac{1}{A} - \frac{1}{B} \right) \]

This integral accounts for the decreasing electric field strength as the distance from the charged shell increases. Substituting this potential difference back into the capacitance formula yields:

\[ C = \frac{Q}{V_{AB}} = \frac{Q}{kQ \left( \frac{1}{A} - \frac{1}{B} \right)} = \frac{1}{k} \frac{AB}{B - A} \]

Expressing k in terms of the permittivity of free space, the capacitance becomes:

\[ C = 4 \pi \varepsilon_0 \frac{AB}{B - A} \]

This formula elegantly captures how the capacitance depends on the radii of the spherical shells and the permittivity of the medium between them. It highlights that as the shells get closer (B approaches A), the capacitance increases, reflecting the stronger interaction between the charges.

In summary, calculating capacitance for non-uniform electric fields involves integrating the electric field to find the potential difference, then applying the fundamental relation C = Q / V. This approach extends beyond spherical capacitors to other geometries like cylindrical capacitors, where the electric field similarly varies with position. Understanding these principles is essential for analyzing real-world capacitors with complex shapes and for designing devices with specific capacitance requirements.

Two long charged coaxial cylinders form a capacitor, one of radius $a$, and the other of radius $b$, where . The inner conductor has charge density $+\lambda$, the outer one with $-\lambda$. Write an expression for the capacitance per length.

• A. $\frac{2\pi {ϵ}_{0_{{}_{}}}}{\ln \left(\frac{b}{a}\right)}$
• B. $2k\lambda \ln \left(\frac{a}{b}\right)$
• C. $\frac{2\pi {ϵ}_{0_{{}_{}}}}{\ln \left(\frac{a}{b}\right)}$
• D. $2\pi {ϵ}_0$

Answer: (C)

Here's what students ask on this topic:

How do you calculate the capacitance of a spherical capacitor using calculus?

To calculate the capacitance of a spherical capacitor, which consists of two concentric spherical conducting shells, we use the relationship $C=\frac{Q}{V}$, where $Q$ is the charge and $V$ is the potential difference between the shells. Since the electric field $E$ is not uniform, we find $V$ by integrating the electric field: $V=-{\int }_A^{B}E\cdot dl$. For spherical shells with radii $a$ and $b$, the electric field behaves like a point charge: $E=\frac{\mathrm{kQ}}{r^2}$. Integrating from $a$ to $b$ gives $V=\mathrm{kQ}(\frac{1}{a}-\frac{1}{b})$. Substituting back, the capacitance is $C=\frac{\mathrm{4C0B5\_0\; ab}}{}$(b - a). This formula shows how geometry affects capacitance when the electric field varies with distance.

Why do we need to use calculus to find the capacitance of non-parallel plate capacitors?

In non-parallel plate capacitors, such as spherical or cylindrical capacitors, the electric field is not uniform because the distance between the charged surfaces varies. Unlike parallel plates where the electric field is constant, here the field strength changes with position, typically decreasing as you move away from the charged surface. To find the potential difference $V$ between the conductors, we cannot simply multiply the electric field by distance. Instead, we use calculus to integrate the electric field along the path between the surfaces: $V=-{\int }_A^{B}E\cdot dl$. This integral accounts for the changing electric field, allowing us to accurately calculate the voltage and thus the capacitance $C=\frac{Q}{V}$. Calculus is essential for these geometries to handle the variable field and distance.

What is the expression for the capacitance per unit length of a cylindrical capacitor?

For a cylindrical capacitor made of two concentric cylindrical shells, the capacitance per unit length is derived similarly to the spherical case but considering the geometry of cylinders. The electric field between the cylinders depends on the radial distance and is given by $E=\frac{\mathrm{kQ}}{r}$, where $r$ is the radius. The potential difference between the inner radius $a$ and outer radius $b$ is found by integrating: $V=-{\int }_a^{b}E\cdot dr$. This results in $V=\frac{\mathrm{kQ}}{\mathrm{2C0B5\_0\; L}}ln(\frac{b}{a})$, where $L$ is the length of the cylinder. The capacitance per unit length is then $\mathrm{C\text{'}}=\frac{\mathrm{2C0B5\_0}}{\mathrm{ln(b/a)}}$. This formula is crucial because an infinitely long cylinder would have infinite total capacitance, so we consider capacitance per unit length.

How does the electric field vary between the plates of a spherical capacitor?

Between the two concentric spherical shells of a spherical capacitor, the electric field behaves like that of a point charge located at the center. It points radially outward and its magnitude decreases with the square of the distance from the center. Mathematically, the electric field at a radius $r$ between the shells is $E=\frac{\mathrm{kQ}}{r^2}$, where $k$ is Coulomb's constant and $Q$ is the charge on the inner shell. This inverse square dependence means the field is strongest near the inner shell and weakens as you move toward the outer shell. This variation requires integration to find the potential difference and capacitance.

What is the general formula for capacitance in terms of charge and potential difference?

The general formula for capacitance is $C=\frac{Q}{V}$, where $C$ is the capacitance, $Q$ is the magnitude of charge stored on the conductors, and $V$ is the potential difference between them. This formula applies to all capacitors regardless of shape. However, for capacitors with non-uniform electric fields, such as spherical or cylindrical capacitors, the potential difference $V$ must be calculated by integrating the electric field along the path between the charged surfaces: $V=-{\int }_A^{B}E\cdot dl$. This integral accounts for the varying electric field, enabling accurate determination of capacitance.