Question 789: Electrostats & Capacitance

Question

A uniformly charged conducting sphere of diameter 3.5 cm has a surface charge density of $20 \mu \mathrm{Cm}^{-2}$. The total electric flux leaving the surface of the sphere is nearly [permittivity of free space, $\varepsilon_0=8.85 	imes 10^{-12} \mathrm{SI}$ unit]

Accepted Answer

Given:
Diameter of sphere, $d = 3.5~	ext{cm} = 3.5 	imes 10^{-2}~	ext{m}$
Surface charge density, $\sigma = 20~\mu	ext{C}/	ext{m}^2 = 20 	imes 10^{-6}~	ext{C}/	ext{m}^2$
Permittivity of free space, $\varepsilon_0 = 8.85 	imes 10^{-12}~	ext{C}^2/	ext{Nm}^2$
Step 1: Formula for Electric Flux (GaussÃ¢ÂÂs Law)
Total electric flux, $\phi$, leaving the surface is:
$$ \phi = \frac{q}{\varepsilon_0} $$
where $q$ is the total charge on the sphere.
Step 2: Calculate Area of the Sphere
Radius, $r = \frac{d}{2} = \frac{3.5}{2}~	ext{cm} = 1.75 	imes 10^{-2}~	ext{m}$
Area, $A$:
$$ A = 4\pi r^2 = 4\pi (1.75 	imes 10^{-2})^2 $$
Calculate:
$$ A = 4 	imes 3.14 	imes (1.75 	imes 10^{-2})^2 \ = 12.56 	imes (3.0625 	imes 10^{-4}) \ = 12.56 	imes 3.0625 	imes 10^{-4} \ \approx 38.48 	imes 10^{-4} \ = 3.848 	imes 10^{-3}~	ext{m}^2 $$
Step 3: Find Total Charge
Total charge, $q = \sigma 	imes A$:
$$ q = 20 	imes 10^{-6}~	ext{C/m}^2 	imes 3.848 	imes 10^{-3}~	ext{m}^2 \ = 76.96 	imes 10^{-9}~	ext{C} \ = 7.696 	imes 10^{-8}~	ext{C} $$
Step 4: Calculate the Total Electric Flux
$$ \phi = \frac{q}{\varepsilon_0} = \frac{7.696 	imes 10^{-8}}{8.85 	imes 10^{-12}} $$
Calculate denominator:
$$ \phi = \frac{7.696}{8.85} 	imes 10^{-8+12} \ = 0.870 	imes 10^{4} \ = 8.7 	imes 10^{3}~	ext{Wb} $$
Step 5: Rounding off
Among the given options, the closest is:
$$ 8.7 	imes 10^3~	ext{Wb} \approx 87 	imes 10^2~	ext{Wb} $$
Final Answer:
Option C: $87 	imes 10^2~	ext{Wb}$

Answer

$57 	imes 10^2 \mathrm{~Wb}$

Answer

$70 	imes 10^2 \mathrm{~Wb}$

Answer

$87 	imes 10^2 \mathrm{~Wb}$

Answer

$35 	imes 10^3 \mathrm{~Wb}$

Electrostats & Capacitance

A uniformly charged conducting sphere of diameter 3.5 cm has a surface charge density of 20 \mu \mathrm{Cm}^{-2}. The total electric flux leaving the surface of the sphere is nearly [permittivity of free space, \varepsilon_0=8.85 \times 10^{-12} \mathrm{SI} unit]

Choose the correct answer: