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Charra [1.4K]
3 years ago
11

An insulating hollow sphere has inner radius a and outer radius b. Within the insulating material the volume charge density is g

iven by rho(r)=αr,where α is a positive constant.
A). What is the magnitude of the electric field at a distance r from the center of the shell, where a Express your answer in terms of the variables α, a, r, and electric constant ϵ0.
B) .A point charge
q is placed at the center of the hollow space, at r=0. What value must q have (sign and magnitude) in order for the electric field to be constant in the region a Express your answer in terms of the variables α, a, and appropriate constants.
C). What then is the value of the constant field in this region?
Express your answer in terms of the variable αand electric constant ϵ0.
Physics
1 answer:
allsm [11]3 years ago
4 0

Answer:

E   = α/2∈₀ [ 1 - a²/r² ]

Ф = α/2∈₀

Explanation:

Using Gauss Law:

    ρ(r) = a/r, dA

          = 4 π r²d r

    Ф = \int\limits^r_a ρ(r')dA

    Ф_{encl} = \int\limits^r_a ρ(r')dA

             = 4πα \int\limits^r_a r'dr'

Ф_{encl}     = 4 π α 1/2(r²-a²)

E(4πr²) = 2\pi\alpha  (r^{2}-a^{2}  )/∈₀

           = 2\pi\alpha  (r^{2}-a^{2}  )/∈₀(4πr²)

           = α (r² - a²) / 2 ∈₀ (r²)

           = α/2∈₀ [ r²/r² - a²/r² ]

      E   = α/2∈₀ [ 1 - a²/r² ]

Electric field of the point charge:

E_{q} = q / 4π∈₀r²

E_{total} = α / 2 ∈₀ - (α / 2 ∈₀ )(a² / r²) + q / 4 π ∈₀ r²

For E_{total}  to be constant:

- (αa²/ 2 ∈₀ ) + q / 4 π ∈₀ = 0 and q = 2παa²

-> α / 2 ∈₀ - αa²/ 2 ∈₀ + 2παa² / 4 π ∈₀

= α - αa² + αa² / 2 ∈₀

= α /2 ∈₀

Hence:

Ф = α/2∈₀

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