Question

Using \(\delta\) functions, write the following mass or charge density functions. (a) Mass 5 at \(x=2,\) and mass 3 at \(x=-7\) (b) Charge 3 at \(x=-5\) and charge -4 at \(x=10\).

Short answer

Mass density: \(\rho(x) = 5 \delta(x - 2) + 3 \delta(x + 7)\). Charge density: \(\rho(x) = 3 \delta(x + 5) - 4 \delta(x - 10)\).

Step-by-step solution

- Understanding \(\backslash\delta\text{ functions}\)

Delta functions, denoted as \(\delta(x - a)\), are used to represent point masses or charges located at specific positions. The expression \(m \delta(x - a)\) represents a mass \(m\) concentrated at position \(a\).

- Setting up mass density for part (a)

To represent a mass of 5 at \(x = 2\), write \(5 \delta(x - 2)\). For a mass of 3 at \(x = -7\), write \(3 \delta(x + 7)\). Combine these to form the total mass density function: \(\rho(x) = 5 \delta(x - 2) + 3 \delta(x + 7)\).

- Setting up charge density for part (b)

To represent a charge of 3 at \(x = -5\), write \(3 \delta(x + 5)\). For a charge of -4 at \(x = 10\), write \(-4 \delta(x - 10)\). Combine these to form the total charge density function: \(\rho(x) = 3 \delta(x + 5) - 4 \delta(x - 10)\).

Key concepts

mass density

Mass density is a measure of how much mass is concentrated in a specific region. It is typically represented as a function in space.

When dealing with point masses, delta functions \([\delta(x - a)]\) come in handy. They are able to represent mass concentrated at a single point. A point mass of 5 at \(x = 2\) would be written as 5\(\delta(x - 2)\). Similarly, a point mass of 3 at \(x = -7\) is represented by \(3 \delta(x + 7)\).

To combine multiple point masses into one function, simply add their delta function representations. For example, having a mass of 5 at \(x = 2\) and a mass of 3 at \(x = -7\), the total mass density function becomes:

\[ \rho(x) = 5 \delta(x - 2) + 3 \delta(x + 7) \].

This function summarizes the mass distribution across the line, with the delta functions pinpointing the exact locations and magnitudes of mass.

charge density

Charge density describes how electric charge is distributed in space. Like mass density, it can involve point charges.

The delta function \(\delta(x - a)\) can represent charges as well. For example, a charge of 3 at \(x = -5\) is shown as \(3 \delta(x + 5)\). If there is a negative charge, such as -4 at \(x = 10\), it is written as \(-4 \delta(x - 10)\).

To represent the total charge density, add the delta functions of each charge together. Thus, with charge 3 at \(x = -5\) and charge -4 at \(x = 10\), the formula becomes:

\[ \rho(x) = 3 \delta(x + 5) - 4 \delta(x - 10) \]

This function outlines the distribution of charges, showing where and how much charge is present at specific points on the line.

point mass

A point mass is an idealized model where mass is assumed to be concentrated at a single point in space.

Even though in reality, mass is distributed over a volume, point masses simplify calculations for many physics problems. Delta functions \(\delta(x - a)\) are incredibly useful in representing these point masses mathematically.

If you have a point mass \(m\) located at position \(a\), it can be denoted as \(m \delta(x - a)\). This notation reflects both the amount of mass \(m\) and its specific location \(a\).

For example, a point mass of 5 at \(x = 2\) is shown as \(5 \delta(x - 2)\). This notation captures the essence of a point mass and is extremely useful in problems involving mass density and distributions.