find the eqivalent capacitance between A and B
Answers
Answer:
- The equivalent capacitance (C_{eq}) is 55C/111
Explanation:
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Firstly Refer the attachment.
Let's consider Δ RST
we can make out that capacitors in arm RS and ST are in series connection. (Same charge will flow through them), let their equivalent capacity be C'
As they are connected in series,
⇒ 1/C' = 1/C₁ + 1/C₂
Substituting the values,
⇒ 1/C' = 1/C + 1/C
⇒ 1/C' = (1 + 1) / C
⇒ 1/C' = 2/C
⇒ C' = C/2
Now, this equivalent capacity (C') is parallel with the capacitor present in TR arm.
Therefore,
⇒ C" = C₃ + C'
Substituting the values,
⇒ C" = C + C/2
⇒ C" = (2 C + C) / 2
⇒ C" = 3 C/2
⇒ C" = 3 C/2
∴ Equivalent capacity in TR arm is 3 C/2.
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Now, lets consider Δ QRT
(now, we need to neglect RS and ST arms as we have solved it)
From diagram we can make out that Capacitors in QR and RT are in series connection.
From the formula we know,
⇒ 1/C'" = 1/C" + 1/C₄
Substituting the values,
⇒ 1/C'" = 2/3C + 1/C
⇒ 1/C'" = (2+3)/3C
⇒ 1/C'" = 5/3C
⇒ C'" = 3C/5
Now, this equivalent capacity (C'") is parallel with the capacitor present in
QT arm.
⇒ C"" = C"' + C₅
Substituting the values,
⇒ C"" = 3C/5 + C
⇒ C"" = (3C + 5C)/5
⇒ C"" = 8C/5
∴ Equivalent capacity in QT arm is 8 C/5.
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Now, lets consider Δ QTU
we can make out that capacitors in arm TU and QT are in series connection. (Same charge will flow through them), let their equivalent capacity be Cₓ
As they are connected in series,
⇒ 1/Cₓ = 1/C"" + 1/C₆
Substituting the values,
⇒ 1/Cₓ = 5/8C + 1/C
⇒ 1/Cₓ = (5 + 8) / 8C
⇒ 1/Cₓ = 13/8C
⇒ Cₓ = 8C/13
Now, this equivalent capacity (Cₓₓ) is parallel with the capacitor present in QU arm.
Therefore,
⇒ Cₓₓ = C₇ + Cₓ
Substituting the values,
⇒ Cₓₓ = C + 8C/13
⇒ Cₓₓ = (13C + 8C) / 13
⇒ Cₓₓ = 21 C/13
⇒ Cₓₓ = 21 C/13
∴ Equivalent capacity in QU arm is 21 C/13.
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Now, lets consider Δ PQU
From diagram we can make out that Capacitors in PQ and QU are in series connection.
From the formula we know,
⇒ 1/Cₐ = 1/Cₓₓ + 1/C₈
Substituting the values,
⇒ 1/Cₐ = 13/21C + 1/C
⇒ 1/Cₐ = (13 + 21)/21C
⇒ 1/Cₐ = 34/21C
⇒ Cₐ = 21 C/34
Now, this equivalent capacity (Cₐₐ) is parallel with the capacitor present in
PU arm.
⇒ Cₐₐ = Cₐ + C₉
Substituting the values,
⇒ Cₐₐ = 21C/34 + C
⇒ Cₐₐ = (21C + 34C)/34
⇒ Cₐₐ = 55C/34
∴ Equivalent capacity in QT arm is 55 C/34.
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Now, in the arm APUB all are in series connection.
⇒ 1/C(eq) = 1/C₁₀ + 1/Cₐₐ + 1/C₁₁
Substituting the values,
⇒ 1/C(eq) = 1/C + 34/55C + 1/C
⇒ 1/C(eq) = (55 + 1 + 55) / 55C
⇒ 1/C(eq) = 111 / 55C
⇒ C(eq) = 55C/111
∴ The equivalent capacitance (C_{eq}) is 55C/111.
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