Q22P - Fundamentals Of Physics

Question

Density, density, density. (a) A charge -300e is uniformly distributed along a circular arc of radius 4.00 cm, which subtends an angle of 40^o. What is the linear charge density along the arc? (b) A charge -300e is uniformly distributed over one face of a circular disk of 2.00 cm radius. What is the surface charge density over that face? (c) A charge -300e is uniformly distributed over the surface of a sphere of radius 2.00 cm. What is the surface charge density over that surface? (d) A charge -300e is uniformly spread through the volume of a sphere of radius 2.00 cm. What is the volume charge density in that sphere?

Step-by-Step Solution

Answer

1. The linear charge density along the arc is $-1.72\times {10}^{-15} \mathrm{C}/\mathrm{m}$
2. The surface charge density over that face is $-3.82\times {10}^{-14} \mathrm{C}/{\mathrm{m}}^2$ 
3. The surface charge density over that surface is $-9.56\times {10}^{-15} \mathrm{C}/{\mathrm{m}}^2$
4. The volume charge density in that sphere is $-1.43\times {10}^{-12} \mathrm{C}/{\mathrm{m}}^3$

Step 1: The given data

• A charge of -300e is uniformly distributed along a circular disc of radius r=4 cm, which subtends at an angle 40^o.
• A charge of -300e is uniformly distributed over one face of a circular disk of radius r=2 cm
• A charge of -300e is uniformly spread through the volume of a sphere of radius r=2 cm

Step 2: Understanding the concept of electric field

The linear charge density is defined as the charge per unit length along a line.

The surface charge density is defined as the charge per unit surface area of a particular surface. The volume charge density is defined as the charge per unit volume of the object.

Using the concept of linear charge density, surface charge density, and volume charge density, the required values of all the cases can be calculated using the given data. The linear density of the body is given by calculating the arc of the circular disk.

 

Formulae:

The formula of linear charge density, $\mathrm{\lambda }=\mathrm{q}/\mathrm{L}$                                              (i)

The formula of surface charge density, $\mathrm{\sigma }=\mathrm{q}/\mathrm{A}$                                          (ii)

The formula of the volume charge density, $\mathrm{\rho }=\mathrm{q}/\mathrm{V}$                                    (iii)

The length of the arc of a circle, $\mathrm{L}=\mathrm{r\theta }$                                                         (iv)

where, $\mathrm{\theta }$ is the angle value in radians. 

The area of the circle, $\mathrm{A}={\mathrm{\pi r}}^2$                                                                        (v)

The surface area of the sphere,  $4\pi r^2$                                                      (vi)

The volume of the sphere,$\mathrm{V}=4{\mathrm{\pi r}}^3/3$                                                     (vii)

Here, r is the radius of the disk

Step 3: a) Calculation of the linear charge density along the arc

The length of the arc of the circular disk is given using equation (iv) as given:

$\begin{array}{rcl}\mathrm{L}& =& \left(0.0400 \mathrm{m}\right)\left(0.698 \mathrm{rad}\right)\\ & =& 0.0279 \mathrm{m}\end{array}$ 

Substituting the given data and the above value in equation (i), we can get the linear density along the arc as:

data-custom-editor="chemistry" $\begin{array}{rcl}\mathrm{\lambda }& =& -300\left(1.602\times {10}^{-19} \mathrm{C}\right)/\left(0.0279 \mathrm{m}\right)\\ & =& -1.72\times {10}^{-15} \mathrm{C}/\mathrm{m}\end{array}$

Hence, the value of the linear charge density is $-1.72\times {10}^{-19} \mathrm{C}/\mathrm{m}$

Step 4: b) Calculation of the surface charge density over that face

Now, the area of the circular face using equation (v) is given as:

$\begin{array}{rcl}\mathrm{A}& =& \mathrm{\pi }{\left(0.0200 \mathrm{m}\right)}^2\\ & =& 0.001256 {\mathrm{m}}^2\end{array}$

 

 Thus, the surface charge density over that face using equation (ii) is given as: 

$\begin{array}{rcl}\mathrm{\sigma }& =& -300\left(1.602\times {10}^{-19} \mathrm{C}\right)/0.001256 {\mathrm{m}}^2\\ & =& -3.82\times {10}^{-14} \mathrm{C}/{\mathrm{m}}^2\end{array}$

 

Hence, the value of the surface charge density is $-3.82\times {10}^{-14} \mathrm{C}/{\mathrm{m}}^2$

Step 5: c) Calculation of the surface charge density over that surface

Area of the sphere is given using equation (vi) is given as:

 

$\begin{array}{rcl}A& =& 4\mathrm{\pi }{\left(0.0200 \mathrm{m}\right)}^2\\ & =& 0.005024 {\mathrm{m}}^2\end{array}$

 

Thus, the surface charge density over that surface using equation (ii) is given as:

 

$\begin{array}{rcl}\mathrm{\sigma }& =& -300\left(1.602\times {10}^{-19} \mathrm{C}\right)/\left(0.005024 {\mathrm{m}}^2\right)\\ & =& -9.56\times {10}^{-14} \mathrm{C}/{\mathrm{m}}^2\end{array}$

 

Hence, the value of the surface charge density is 

Step 6: d) Calculation of the volume charge density in that surface

The volume of the sphere using equation (vii) is given as:

 

 

Finally, the volume charge density in that surface is given using equation (iii) as:

 

 Hence, the value of the volume charge density is