A coaxial conductor has radii \(a=0.8 \mathrm{~mm}\) and \(b=3 \mathrm{~mm}\) and a polystyrene dielectric for which \(\epsilon_{r}=2.56 .\) If \(\mathbf{P}=(2 / \rho) \mathbf{a}_{\rho} \mathrm{nC} / \mathrm{m}^{2}\) in the dielectric, find (a) \(\mathbf{D}\) and \(\mathbf{E}\) as functions of \(\rho ;(b) V_{a b}\) and \(\chi_{e} \cdot(c)\) If there are \(4 \times 10^{19}\) molecules per cubic meter in the dielectric, find \(\mathbf{p}(\rho)\)

Question: Calculate the electric field (E), electric displacement field (D), potential difference (V_ab), electric susceptibility (χe), and the polarization (P) of a polystyrene dielectric in a coaxial conductor with radii a and b. Solution: 1. Write the formula for the electric displacement field (D) in cylindrical coordinates: D = Dρ(ρ)ar. 2. Express D in terms of polarization P: Dρ(ρ) = ε0Eρ + Pρ. 3. Solve for the electric field (E) as a function of ρ: E(ρ) = (Dρ(ρ) - Pρ) / ε0 ar. 4. Write the formula for potential difference in a coaxial conductor: Vab = -∫[a,b] Eρ(ρ) dρ. 5. Calculate the potential difference V_ab using the expression for E from step 3. 6. Write the formula for electric susceptibility (χe): χe = Pρ / (ε0Eρ). 7. Calculate the electric susceptibility χe using the given values of P and E, and the value of ε0. 8. Write the formula for polarization p(ρ) in terms of the number of molecules per unit volume: p(ρ) = P(ρ) / N. 9. Calculate p(ρ) using the given values of P and N.