Question

27 spherical droplets each of radius 3mm and carrying 10 to the power minu 12c of charge are combined to form a single drop. find the capacitance and potential of the bigger drop

Answer 1

GIVEN:

• Number of droplets, n = 27
• Radius of single droplet, r = 3mm = 3 × 10^-3 m
• Charge of single droplet, q = 10^-12 C

TO FIND:

• Capacitance of bigger drop, C
• Potential of bigger drop, V

FORMULAE:

• Volume of spherical droplets = (4πr³) / 3

• Capacitance, C = 4π$\epsilon_0$r

• Potential, V = Q / C

where, $\epsilon_0$ is the permittivity of free space and is equal to 8.854 × 10^-12 C^2 N^-2 m^-2

SOLUTION:

STEP 1:

• 27 spherical droplets combine to form large single droplet.

• According to law of conservation of charges, total charge remains constant.

• Total charges of individual droplets = Charge of single big droplets

• Q = nq : where Q is the charge of bigger droplet and q is the charge of single small droplet.

Q = nq

Q = 27 × 10^-12 C

STEP 2:

• Let R be the radius of bigger droplet and r be the radius of single small droplet.

• Volume remains constant before and after the combination of droplets.

• Volume of bigger droplet = Total volume of smaller droplets.

(4πR³)/3 = n × (4πr³)/3

R³ = n r³

n = R³/r³

n = (R /r )³

27 = (R / r )³

3³ = (R / r )³

3 = R / r

R = 3r

R = 3 × 3× 10^-3

R = 9 × 10^-3 m

STEP 3:

Let C be the capacitance of bigger droplet .

Capacitance of bigger droplet , C = 4π$\epsilon_0$R

$C = 4 \times 3.14 \times 8.854 \times {10}^{ - 12} \times 9 \times {10}^{ - 3} \\ \\ = 4 \times 3.14 \times 8.854 \times 9 \times {10}^{ - 15} \\ \\ C = 1000.856 \times {10}^{ - 15} \: farad$

STEP 4 :

Let V be the potential of bigger droplet.

$V = \dfrac{Q}{C} \\ \\ V = \dfrac{27 \times {10}^{ - 12} }{1000.856 \times {10}^{ - 15} } \\ \\ V = \dfrac{27}{1000.856} \times {10}^{ - 12} \times {10}^{15} \\ \\ V = 0.0269 \times {10}^{ (- 12 + 15)} \\ \\ V = 0.0269 \times {10}^{3} \\ \\ V = 26.9 \: volt$

ANSWER:

• Capacitance of bigger droplet is 1000.856 × 10^-15 F

• Potential of bigger droplet is 26.9 volt

Answer 2

$\mathbb{ \ \ A\ N\ S\ W\ E\ R\: \: }$

• Capacitance of bigger droplet is 1000.856 × 10-¹⁵ F
• Potential of bigger droplet is 26.9 volt