Question

Using $\mathit{\delta }$functions, write the following mass or charge density functions.

(a) Mass 5 at $\mathit{x}\mathbf{}\mathbf{=}\mathbf{}\mathbf{2}$, and mass 3 at $\mathit{x}\mathbf{}\mathbf{=}\mathbf{-}\mathbf{}\mathbf{7}$.

(b) Charge 3 at $\mathit{x}\mathbf{=}\mathbf{-}\mathbf{5}$and charge -4at$\mathit{x}\mathbf{=}\mathbf{10}$.

Short answer

(a) The mass density function is $5\delta (x-2)+3·\delta (x+7)$.

(b) The charge density function is $3·\delta (x+5)-4·\delta (x-10)$.

Step-by-step solution

Given information

The given expressions are

(a)Mass 5 at x=2, and mass 3 at x=-7.

(b) Charge 3 at x=-5 and charge -4 at x=10.

Definition of Mass Density

The mass density of an object is defined as its mass per unit volume. The formula for density is d = M/V, where d is density, M is mass, and V is volume. Density is commonly expressed in units of grams per cubic centimeter.

(a) Calculate the mass density for given value at x=2, and at x=-7.

The density function for a point mass m at x=a is$m·\delta (x-a)$

Mass$m_1=5$

$a_1=2$

Mass$m_2=3$

$a_2=-7$

Mass density function

$$\begin{gathered} =m_1\delta \left(x-a_1\right)+m_2·\delta \left(x-a_2\right) \\ =5\delta (x-2)+3·\delta (x+7) \end{gathered}$$

Thus, the mass density function is $5\delta (x-2)+3·\delta (x+7)$.

(b) Calculate the mass density for given value at x=-5, and at x=10

The charge function for a point mass m at x=a is$q_1·\delta (x-a)$

Charge$q_1=3$

$a_1=-5$

Mass$q_2=-4$

$a_2=10$

Charge density function

$$\begin{gathered} =q_1\delta \left(x-a_1\right)+q_2·\delta \left(x-a_2\right) \\ =3·\delta (x+5)-4·\delta (x-10) \end{gathered}$$

Thus, the mass density function is $3·\delta (x+5)-4·\delta (x-10)$.