Question 108: Electrostats & Capacitance

Question

Three identical capacitors $${C_1},{C_2}$$ and $${C_3}$$ have a capacitance of $$1.0\,\mu F$$ each and they are unchanged initially. They are connected in a circuit as shown in the figure and $${C_1}$$ is then filled completely with a dielectric  material of relative permittivity $${\varepsilon _r}.$$ The cell electromotive force $$\left( {emf} \right)\,\,{V_0} = 8V.$$ First the switch $${S_1}$$ is closed while the switch $${S_2}$$ is kept open. When the capacitor $${C_3}$$ is fully charged, $${S_1}$$ is opened and $${S_2}$$ is closed simultaneously. When all the capacitors reach equilibrium, the charge on $${C_3}$$ is found to be $$5\,\mu C.$$ The value of $${\varepsilon _r} = $$ _________________.

Accepted Answer

Given C
1
= C
2
= C
3
= 1.0 $$\mu$$F and V
0
= 8 V
When S
1
is close and S
2
is open. Charge on C
3
is 8 $$\mu$$C.
The circuit is
When S
1
is open and S
2
is close. Charge on C
3
is 5 $$\mu$$C. Therefore, charge on C
1
and C
2
will be 3 $$\mu$$C.
The potential difference across C
3
is equal to the potential difference across C
1
and C
2
together i.e.,
$${{{q_1}} \over {C{'_1}}} + {{{q_2}} \over {{C_2}}} = {{{q_3}} \over {{C_3}}}$$,
$${{3\mu C} \over {{ \in _r}(1\mu F)}} + {{3\mu C} \over {1\mu F}} = {{5\mu C} \over {1\mu F}}$$.
Solve to get $${ \in _r} = 1.5$$.

Electrostats & Capacitance

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