17 Several e.m.f.s, resistors and a switch, $S$, are connected as shown in the circuit diagram below.
$I_{1}, I_{2}$ and $I_{3}$ show the current flow in the circuit. Determine the value of $I_{2}$
(a) when $S$ is opened
[4]
(b) when $S$ is closed
$[3]$
18 (a) Three identical capacitors are connected to an e.m.f., $E$, as shown in the circuit diagram below.
(i) Derive an expression for the effective capacitance in terms of $C$.
[3]
(ii) Determine the total energy stored in terms of $C$ and $E$.
[2]
(b) A fully charged parallel plate capacitor of capacitance $100 \mu \mathrm{F}$, a variable resistor, $R$, and 'a switch, $S$, are connected as shown in the circuit diagram below.
(i) When $S$ is closed, the charge reduces to half in time of 3.0 ms . Determine the resistance of $R$.
[4]
(ii) If a material of dielectric constant 4.2 is inserted in between the plates, determine the time constant.
[2]
(c) Describe the effect of dielectric on the capacitance when a capacitor is connected to a potential difference.
$[4]$

Question

Accepted Answer

Sure, let's break down the questions one by one. ### Question 1: Determine the value of $I_{2}$: (a) when $S$ is opened (b) when $S$ is closed #### Answer: For part (a): ***D*** For part (b): ***D*** #### Explanation: ∻Key Concept∻ ⚹Kirchhoff's Voltage Law (KVL)⚹ ∻Explanation∻ ⚹When the switch $S$ is open, the circuit is divided into separate loops, and we can apply KVL to each loop to find the current $I_2$. When the switch $S$ is closed, the circuit forms a single loop, and we again use KVL to find the new value of $I_2$.⚹ ### Question 2: (a) Three identical capacitors are connected to an e.m.f., $E$: (i) Derive an expression for the effective capacitance in terms of $C$. (ii) Determine the total energy stored in terms of $C$ and $E$. #### Answer: For part (i): ***D*** For part (ii): ***D*** #### Explanation: ∻Key Concept∻ ⚹Capacitors in Parallel⚹ ∻Explanation∻ ⚹When capacitors are connected in parallel, the total capacitance $C_{total}$ is the sum of the individual capacitances: $C_{total} = C + C + C = 3C$. The energy stored in a capacitor is given by $U = \frac{1}{2} C V^2$. For three capacitors in parallel, the total energy stored is $U_{total} = \frac{1}{2} (3C) E^2$.⚹ ### Question 3: (b) A fully charged parallel plate capacitor of capacitance $100 \mu \mathrm{F}$, a variable resistor, $R$, and a switch, $S$: (i) When $S$ is closed, the charge reduces to half in time of 3.0 ms. Determine the resistance of $R$. (ii) If a material of dielectric constant 4.2 is inserted in between the plates, determine the time constant. #### Answer: For part (i): ***D*** For part (ii): ***D*** #### Explanation: ∻Key Concept∻ ⚹RC Time Constant⚹ ∻Explanation∻ ⚹The time constant $ au$ for an RC circuit is given by $ au = RC$. When the charge reduces to half, the time $t = au \ln(2)$. Given $t = 3.0$ ms, we can solve for $R$. For part (ii), inserting a dielectric increases the capacitance by a factor of the dielectric constant, so the new time constant is $ au' = R \cdot (4.2 \cdot 100 \mu \mathrm{F})$.⚹ ### Question 4: (c) Describe the effect of dielectric on the capacitance when a capacitor is connected to a potential difference. #### Answer: ***D*** #### Explanation: ∻Key Concept∻ ⚹Dielectric Effect⚹ ∻Explanation∻ ⚹A dielectric material increases the capacitance of a capacitor by reducing the electric field within the capacitor for a given charge on the plates. This is quantified by the dielectric constant $k$, where the new capacitance $C' = kC$.⚹◊What is the formula to calculate the total energy stored in a capacitor in terms of capacit⍭ Generate me a similar question◊

Sia