Dielectric Sphere Electric Field Calculation

The electric field ($E_{center}$) at the center of a uniformly polarized dielectric sphere depends on the polarization ($P$) and the permittivity of free space ($\epsilon_0$). The formula for the electric field at the center is:

$ E_{center} = -\frac{P}{3\epsilon_0} $

The question asks for the magnitude of the electric field, denoted as $E \times 10^9 \text{ N. C}^{-1}$. We calculate the magnitude:

$ |E_{center}| = \frac{P}{3\epsilon_0} $

Unit Conversion

First, convert the polarization $P$ from $\mu \text{C.cm}^{-2}$ to SI units (C.m$^{-2}$).

• Given: $P = 26 \, \mu \text{C.cm}^{-2}$
• Conversion factors: $1 \, \mu \text{C} = 10^{-6} \, \text{C}$ and $1 \, \text{cm} = 10^{-2} \, \text{m}$.
• $1 \, \text{cm}^{-2} = (10^{-2} \, \text{m})^{-2} = 10^4 \, \text{m}^{-2}$.
• So, $P = 26 \times 10^{-6} \, \text{C} \times 10^4 \, \text{m}^{-2} = 26 \times 10^{-2} \, \text{C.m}^{-2} = 0.26 \, \text{C.m}^{-2}$.

Magnitude Calculation

Now substitute the converted polarization and the given $\epsilon_0$ into the magnitude formula:

• $P = 0.26 \, \text{C.m}^{-2}$
• $\epsilon_0 = 8.85 \times 10^{-12} \, \text{C}^2\text{.N}^{-1}\text{.m}^{-2}$
• $ |E_{center}| = \frac{0.26 \, \text{C.m}^{-2}}{3 \times (8.85 \times 10^{-12} \, \text{C}^2\text{.N}^{-1}\text{.m}^{-2})} $
• $ |E_{center}| = \frac{0.26}{26.55 \times 10^{-12}} \, \text{N.C}^{-1} $
• $ |E_{center}| \approx 0.0097928 \times 10^{12} \, \text{N.C}^{-1} $
• $ |E_{center}| \approx 9.7928 \times 10^9 \, \text{N.C}^{-1} $

Determining E

The electric field magnitude is given in the form $E \times 10^9 \text{ N.C}^{-1}$. Comparing this with the calculated value:

• $E \times 10^9 \approx 9.7928 \times 10^9$
• $E \approx 9.7928$

Rounding off to one decimal place, we get $E = 9.8$.