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

Short Answer

Expert verified
  1. The linear charge density along the arc is-1.72×10-15C/m
  2. The surface charge density over that face is-3.82×10-14C/m2
  3. The surface charge density over that surface is-9.56×10-15C/m2
  4. The volume charge density in that sphere is-1.43×10-12C/m3

Step by step solution

01

The given data

  • A charge of -300e is uniformly distributed along a circular disc of radius r=4 cm, which subtends at an angle 40o.
  • 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
02

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,λ=q/L (i)

The formula of surface charge density,σ=q/A (ii)

The formula of the volume charge density,ρ=q/V (iii)

The length of the arc of a circle,L= (iv)

where,θis the angle value in radians.

The area of the circle,A=πr2 (v)

The surface area of the sphere, 4πr2 (vi)

The volume of the sphere,localid="1663926381499" V=4πr3/3 (vii)

Here, r is the radius of the disk

03

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:

L=0.0400m0.698rad=0.0279m

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" λ=-3001.602×10-19C/0.0279m=-1.72×10-15C/m

Hence, the value of the linear charge density is-1.72×10-19C/m

04

b) Calculation of the surface charge density over that face

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

A=π0.0200m2=0.001256m2

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

σ=-3001.602×10-19C/0.001256m2=-3.82×10-14C/m2

Hence, the value of the surface charge density is-3.82×10-14C/m2

05

c) Calculation of the surface charge density over that surface

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

A=4π0.0200m2=0.005024m2

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

σ=-3001.602×10-19C/0.005024m2=-9.56×10-14C/m2

Hence, the value of the surface charge density is

06

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

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