Problem 23 - Fundamentals of Physics

Question

(a) The drum of a photocopying machine has a length of \(42 \mathrm{~cm}\) and a diameter of \(12 \mathrm{~cm} .\) The electric field just above the drum's surface is \(2.3 \times 10^{5} \mathrm{~N} / \mathrm{C}\). What is the total charge on the drum? (b) The manufacturer wishes to produce a desktop version of the machine. This requires reducing the drum length to \(28 \mathrm{~cm}\) and the diameter to \(8.0 \mathrm{~cm} .\) The electric field at the drum surface must not change. What must be the charge on this new drum?

Step-by-Step Solution

Answer

The charge on the original drum is 43.5 µC. The charge on the new drum is 20.7 µC.

Step 1: Find the surface area of the original drum

The drum can be modeled as a cylinder. The surface area of a cylinder without the ends is calculated using the formula: \( A = \pi d h \) where \( d \) is the diameter and \( h \) is the height (or length) of the cylinder. For the original drum, \( d = 12 \mathrm{~cm} \) and \( h = 42 \mathrm{~cm} \). Therefore, \( A = \pi \times 12 \times 42 \).

Step 2: Calculate the total charge on the original drum

The electric field \( E \) and charge \( Q \) are related by the formula for the surface charge density \( \sigma = \frac{Q}{A} = \varepsilon_0 E \), where \( \varepsilon_0 = 8.85 \times 10^{-12} \mathrm{~C}^2/\mathrm{N} \cdot \mathrm{m}^2 \) is the permittivity of free space. Rearranging gives: \( Q = \varepsilon_0 E A \). Substitute \( E = 2.3 \times 10^{5} \mathrm{~N} / \mathrm{C} \) and \( A = \pi \times 12 \times 42 \), to solve for \( Q \).

Step 3: Find the surface area of the new drum

The new drum also follows the same surface area formula. Here, \( d = 8 \mathrm{~cm} \) and \( h = 28 \mathrm{~cm} \). Thus, the new surface area is \( A_{new} = \pi \times 8 \times 28 \).

Step 4: Calculate the charge on the new drum

Using the same relationship \( Q_{new} = \varepsilon_0 E A_{new} \). Substitute the unchanged electric field \( E = 2.3 \times 10^{5} \mathrm{~N} / \mathrm{C} \) and the new surface area \( A_{new} = \pi \times 8 \times 28 \), calculate \( Q_{new} \).

Key Concepts

Surface Charge Density

Surface charge density is a measure of how much electric charge is accumulated over a given surface area. In physics, it's commonly represented by the symbol \( \sigma \). To find the surface charge density, we use the formula:

• \( \sigma = \frac{Q}{A} \)

where \( Q \) is the total charge and \( A \) is the surface area. This formula essentially tells us how spread out the charge is over the surface. If the charge \( Q \) is high and the area \( A \) is small, the surface charge density will be high, meaning the electric field will be strong in that area.
In the problem of the photocopying machine's drum, the electric field \( E \) is related to the surface charge density by the equation:

• \( \sigma = \varepsilon_0 E \)

where \( \varepsilon_0 \) is the permittivity of free space. This indicates that the electric field is directly proportional to how much charge is on the surface compared to its area.

Cylinder Surface Area

The surface area of a cylinder, excluding the ends, is an important factor in calculating other properties like total charge distributed over it. The surface area \( A \) is given by the formula:

• \( A = \pi d h \)

where \( d \) is the diameter and \( h \) is the height of the cylinder. Imagine the drum of the photocopying machine as a cylindrical surface that wraps around. To understand how much surface area you have, visualize this as unrolling the curved part of a can. The original drum’s surface area is found using its diameter (12 cm) and its height (42 cm). This calculates to \( \pi \times 12 \times 42 \).
For the new, smaller drum, the same cylinder surface area formula is applied, using the diameter of 8 cm and the height of 28 cm, resulting in \( \pi \times 8 \times 28 \). Understanding this concept is crucial to finding out other physical properties like charge on the drum.

Permittivity of Free Space

Permittivity of free space is a fundamental constant denoted by \( \varepsilon_0 \). It quantifies how much resistance is encountered when forming an electric field in a vacuum. In simpler terms, it’s a measure of the ability of the vacuum to allow electric field lines to flow through.
The standard value of permittivity of free space is \( \varepsilon_0 = 8.85 \times 10^{-12} \mathrm{~C}^2/\mathrm{N} \cdot \mathrm{m}^2 \). This constant plays a role in the relationship between electric field and charge distribution given by:

• \( Q = \varepsilon_0 E A \)

Here, \( E \) is the electric field, \( A \) is the surface area, and \( Q \) is the charge. This formula helps us to find out the total charge on the drum by combining these different physical aspects. In electric fields, \( \varepsilon_0 \) acts to moderate how strong the field can be, linking the concepts of charge and surface area seamlessly.