Question

The potential (in volts) of a charge distribution is given by V(z)=30−5z2 for |z|≤1mV(z)=35−10|z| for |z|≥1m V(z) does not depend on x and y .If this potential is generated by a constant charge per unit volume ρ0, (in units of ∈0) which is spread over certain region ,then choose the correct statement

Answer 1

Given:

V(z)=30−5z² for ∣z∣≤1m

V(z)=35−10∣z∣ for ∣z∣≥1m

To find:

The correct statement specifying ρ.

Solution:

• As per the question, V(z)=30−5z² for ∣z∣≤1m and V(z)=35−10∣z∣ for ∣z∣≥1m.

• Thus, the electric field will be E(z)=10z ; ∣z∣≤1m and E(z)=10 ; ∣z∣≥1m.

• Also, for ∣z∣≤1m, E(z) is proportional to z, i.e. continuous charge distribution for  ∣z∣≤1m

• On the other hand , ∣z∣≥1m, E(z) is constant because of the electric field due to an infinite sheet outside it.

• Now, applying Gauss's law for ∣z∣≤1m, we get -

       E x 2A= ρ x 2z x A​ / ϵ₀

              ρ = 10ϵ₀

• And, for ∣z∣≥1m, ρ = 0.

• The correct statement is : ρ = 10ϵ₀ , For ∣ z ∣ ≥ 1m , ρ = 0