Question

A solid cylinder of radius 1.5 m has a uniform volume charge density of 15 C/m³. Find the magnitude of the electric field at 1.25 m from the axis of the cylinder.

a) what will your gaussian surface be? Make a sketch of the solid cylinder and the gaussian surface with their radii

b) Write an expression for the total electric flux through the gaussian surface, that is the LHS (Left hand side) of the Gauss' law (this expression may contrain a quantity you may need to define but you don't know its value).

c) Write the total charge enclosed by the Gaussian surface divided by the dielectric constant (this expression may contain a quantity you may need to define by you don't know its value)

d) use Gauss' law to obtain the final expression and numeric value (with units) for the required electric field.

Answer 1

E_r= (1/4π ϵ0)(Q_inside r/R³)

=(1/4π ϵ0)(ρV/R³)r

E_r(1.25)= =(1/4π ϵ0)x(1.997x10^-6)x15x1.25) /(1.5)³)

Where =ϵ0 =8.854x10^-12

=(262.10625x10^-12x10^-6)/2629.638=0.09967x10⁶

a)For r<R

E=λr/(2πϵ₀R²)

where λ-charge per unit length

b)

Total electric flux through the gaussian surface is   ϕ =Eds

Charge elclosed by the closed surface os q= σ ds

ϕ =q/ϵ0          =(1/ϵ₀)σ ds so Eds=(1/ϵ₀) σ ds

E=σ/ϵ0

C) Gauss law is defined as the total charge Q enclosed within a surface divided by dielectric constant.

ϕ = Q/ϵ0

Where, ϕ = Electric flux through a given surface, Q = total charge within a given surface,

ϵ0 = Electric constant.

the electric flux flowing outwards to the surface is proportional to the total electric charge enclosed by it.So, total charge Q=φε₀

d) Gauss' law is a tool for the calculation of electric fields when they originate from charge distributions of sufficient symmetry to apply it

If the charge distribution lacks sufficient symmetry for the application of Gauss' law, then the field must be found by summing the point charge fields of individual charge elements.

Φ_electric =Q/ ϵ0