(a) Find the potential difference between `A` and `B`

(b) A circuit has a section `AB` as shown. If `V_A - V_B = 12 V`, find current `i`.

( c) (i) Find the potential difference between the points `A` and `B`
(ii) Find current through `25 V` cell if points `A` and `B` are connected.

(d) In the following circuit, calculate
(i) Potential difference between points `X` and `Y`, when switch `S` is open.
(ii) Current through switch `S`, when switch `S` is closed.
(iii) Equivalent resistance between `A` and `B`. `.

Explanation

(a) Move from `A` to `B` :
`V_A - 15 - 5 xx 2 - 10 - 5 xx 6 - 5 = V_B`
`V_A - V_B = 70 V`

(b) `V_A - i xx 3 + 5 - i xx 6 +10 = V_B`
`V_A - V_B + 15 = 9 i`
`12 + 15 = 9 i rArr i = 3 A`

(i)
No current flows in the branch `CB`.
Let current in the curcuit `A X Y C Z W A` be `i`
`A` to `A`, path `A X Y C Z W A`
`V_A - i xx 1 - 25 - i xx 3 - i xx 2 - i xx 1 - i xx 1 + 15 - i xx 2 = V_A`
`10 i = -10 rArr i = -1 A`
i.e., direction of `i` is oppsite.
`A` to `B`, path `A X Y C B`
`V_A - i xx 1 - 25 - i xx 3 - i xx 2 + 10 = V_B`
`V_A - V_B = 15 + 6 i = 15 + 6 xx (-1) = 9 V`
(ii)
`E = ((25)/(6) +(1)/(2) + (15)/(4))/((1)/(6) +(1)/(2)+(1)/(4)) = ((50 + 60 + 45)/(12))/((2 + 6 + 3)/(12)) = (155)/(11) V`
`V_A - V_B = (155)/(11) V`
Cell of emf `25 V`
`V_A - V_B = 25 - 6 i_1`
`(155)/(11) = 25 - 6 i_1`
`6 i_1 = 25 -(155)/(11) =(120)/(11)`
`i_1 = (20)/(11) A`.
(d) (i) `p.d` between `X` and `B`, `V_X - V_B = (2)/(1 + 2) xx 25 = (50)/(3) V`
`p.d`. between `Y` and `B`, `V_Y - V_B = (4)/(3 +4) xx 25 = (100)/(7) V`
`V_X - V_Y = (50)/(3) - (100)/(7) = (50)/(21) V`
(ii)
`X` and `Y` are at same potential
`R_(eq) = (3)/(4)+(4)/(3) = (25)/(12) Omega`
`i = (25)/(25//12) = 12 A`

`i_1 = (3)/(1 + 3) i = (3)/(4) xx 12 = 9 A`
`i_2 = (4)/(2 + 4) i = (2)/(3) xx 12 = 8 A`
Current through `XY` or switch `S`
=`9 - 8 = 1 A`, from `X` to `Y`
(iii) `(R_(eq))_(AB) = (25)/(12) Omega`.