A hollow, thin-walled insulating cylinder of radius R and...

A hollow, thin-walled insulating cylinder of radius $R$ and length $L$ (like the cardboard tube in a roll of toilet paper) has charge $Q$ uniformly distributed over its surface. (a) Calculate the electric potential at all points along the axis of the tube. Take the origin to be at the center of the tube, and take the potential to be zero at infinity. (b) Show that if $L \ll R,$ the result of part (a) reduces to the potential on the axis of a ring of charge of radius $R$. (See Example 23.11 in Section 23.3.) (c) Use the result of part (a) to find the electric field at all points along the axis of the tube.

Key Concepts

Limiting Case Approximations

Limiting case approximations are employed to simplify complex expressions by considering the behavior of the system under certain parameter limits (e.g., one dimension being much smaller than another). This method helps in verifying the consistency of a general result with known results in simpler scenarios or determining the dominant factors in physical behavior when one parameter is significantly small or large compared to another.

Electric Potential from a Continuous Charge Distribution

This concept involves calculating the electric potential by summing (integrating) the contributions from each infinitesimal element of charge in a distribution. The potential is a scalar quantity and is determined by integrating the charge density over the geometry of the system, taking into account the distance from each charge element to the observation point and any symmetry present in the setup.

Electric Field from Electric Potential

In electrostatics, the electric field is derived as the negative gradient of the electric potential. This relationship connects a scalar field (potential) to a vector field (electric field) and is fundamental in determining the force experienced by charges in space. Calculating the potential first and then differentiating it is often an easier route than directly calculating the field.

Superposition Principle

The superposition principle is a key idea in electrostatics which posits that the net electric potential or electric field due to multiple charge contributions is simply the algebraic sum of the potentials or fields from each individual charge distribution. This principle is particularly useful in complex geometries where the overall potential or field can be decomposed into simpler, more manageable parts.

Integration in Cylindrical Coordinates

Utilizing cylindrical coordinates is essential when dealing with systems that exhibit rotational symmetry around an axis, such as cylindrical or ring-like charge distributions. This coordinate system simplifies the integration process by aligning the mathematical description with the symmetry of the physical system, making it easier to set up and evaluate the integrals describing the potential or field.

Transcript

00:01 Welcome to problem number 23 .82 of chapter 23 electric potential.

00:08 So in this question we have a hollow thin -wold insulating cylinder of radius r having length l.

00:18 So the length is l and radius is r.

00:26 Saying thin -wolds insulating cylinder means that whatever the charge is on this material stays there and the width of this cylinder is very very small.

00:37 So you can consider it as a sheet or a very very thin sheet of a cylinder in shape.

00:45 So this has charged q on this way, on this cylinder.

00:54 Now we need to find out the electric potential at some point which is at distance x from the origin.

01:03 And we have placed our cylinder in the midway of the origin.

01:11 So this is our origin.

01:13 So this is our l by 2 and the remaining is also l by 2 and we need to find out the potential along the same axis of the cylinder axis all right so first thing we have to take some element here so let's take some element here so that element is in circle shape and let's say we have this element is at distance z having width equals to d z so this is our reason this is at distance z and we are finding out the potential at x position from the origin.

02:06 So let's say we have dq charge on this one.

02:11 So potential due to this dq at this position is equal to dv equals to 1 by 4 pi a epsilon not.

02:22 The distance from this point to this point is x minus z.

02:29 So the distance is this one is equal to r square plus x minus z whole square.

02:39 Since the radius is same, this radius r which is the radius same...

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