13. CURRENT ELECTRICITY

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Muriel Brooks
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1 . CUNT CTICITY. ce of e.m.f. and interna resistance Ω is connected to two resistances of 0 Ω and Ω joined in parae. Find the current through each resistance using Kirchhoff s aws. Formua : r Ω 0 Ω Ω I Current through?, I Current through?, I Tota current through network i) ΣI 0 (Kirchhoff s Junction aw) ii) ΣI + Σ 0 (Kirchhoff s votage aw) Soution : From Kirchhoff s junction aw, I I I 0 I I + I K I r Ω vot M I ppy Kirchhoff s votage aw for oop in the circuit containing KMNOPQK I Ir + 0 I (I + I ) r I (I + I ) I I I + 0 I I + 0 Current ectricity I 0 Ω N N O Ω Q P I + I... (i) ppy Kirchhoff s votage aw for oop MN ONM I + I 0 I + 0 I 0 I. I... (ii) From equations (i) and (ii) (. I ) + I.6 I + I 7.6 I I 7.6 I (iii) From equations (ii) and (iii) I (iv) I I + I I 0.9 I I 0.7. The current fowing through an externa resistance of Ω is 0., when it is connected to the terminas of a ce. This current reduces to 0. when the externa resistance is Ω. Use Kirchhoff s aws to find e.m.f of ce. Ω Ω I 0., I 0.?

2 MHSH TUTOIS SCINC.. 7 Formua : ΣI + Σ 0 Soution : r I 0. I. Four resistances Ω, 0 Ω, Ω and an unknown Ω, are connected in series so as to form Wheatstone s network. Determine the unknown resistance, if the network is baanced with these numerica vaues of resistances. B Ω 0 Ω (b) et r be the interna resistance of ce. ppy Kirchhoff s votage aw to figure (a) I ( + r) + 0 I ( + r) 0. ( + r)... (i) ppy Kirchhoff s votage aw to figure (b) I ( + r) + 0 I ( + r) 0. ( + r) From equations (i) and (ii) 0. ( + r) 0. ( + r) ( + r) ( + r) + r + r... (ii) r Ω... (iii) I From equations (i) and (iii) 0. ( + ). Ω (a) r Ω. vot 0. I Ω 0 Ω Ω? Formua : Soution : Ω Ω D. In a meter-bridge experiment with resistance in eft and resistance in right gap, nu point is obtained at 0 cm from eft end. With a resistance in eft gap and same resistance in right gap, nu point is obtained at 0 cm from eft end. Where wi be nu point if and are put first in series and then in parae, in the eft gap and right gap sti containing? C Ω Current ectricity

3 .. 76 MHSH TUTOIS SCINC Soution : When is in eft gap and is in right gap, nu point S Current ectricity 0 cm When is in eft gap and in right gap, nu point 0 cm For and in series with eft gap, nu point S? For and in parae in eft gap, nu point P? In baanced position For first case, 0 cm cm 60 cm For second case, 0 cm cm (i) (ii) When and are connected in series, the equivaent resistance is given by S (iii) The nu point in series combination is obtained at S from eft end ( in right gap and series combination in eft gap) 00 S S S S 00 S S 00 (00 S ) S 00 S S 00 8 S S S 00 8 S 6. cm [From (iii)] When and are connected in parae the equivaent resistance is given by P +. + P...(iv) The nu point in parae combination is obtained at P from the eft end ( in right gap and parae combination in eft gap) 00 P P P P 00 P

4 MHSH TUTOIS SCINC potentiometer wire has a ength of m and resistance of 0 Ω. It is connected in series with resistance 990 Ω and a ce of e.m.f. Cacuate the potentia aong the wire. m 0 Ω 990 Ω K? Formua : K Soution : Since I so + I vot K K 0 K 0 /m P 00 (00 P ) P 7 P 00 P 00 7 P 8.6 cm P Two ces having unknown e.m.f and ( > ) are connected in potentiometer circuit so as to assist each other. The nu point is obtained at 8. m from the higher potentia end. When ce is connected so as to oppose ce, the nu point is obtained at. m from same end. Compare the e.m.f s of two ces. 8. m. m Formua : Soution :? potentiometer wire has a ength 0 m and a resistance 0 Ω. Its terminas are connected to a battery of e.m.f and interna resistance Ω. What are the distances at which nu points are obtained when two ces of e.m.f. and. are connected so as to (i) assist and (ii) oppose each other? 0 m 0 Ω Interna resistance of ce, r Ω Current ectricity

5 .. 78 MHSH TUTOIS SCINC. and. Formua : i) Nu point when and are connected in assist? ii) Nu point when and are connected in oppose? i) + K ii) K Soution : i) Since, K K ( + r) I ( + r) K ( ) K 0. /m When and are connected so as to assist then we have i) + K + K m potentiometer wire of ength m and resistance 8 Ω is connected in series with a battery of e.m.f and negigibe interna resistance. If the e.m.f of ce baances against ength of 7 cm of the wire, find the e.m.f. of ce. When a ce is shunted by a resistance of Ω, the baancing ength is reduced by 7 cm. Find the interna resistance of ce. m 8 Ω r 0 7 cm.7 m Ω (7 7) 00 cm m? r? Formua : i) K ii) r Soution : Since, K ( + r) m I ii) K Current ectricity + K.. 0. K K ( +r) ( ) K 0. /m 8 8

6 MHSH TUTOIS SCINC votmeter has a resistance of 00 Ω. What wi be its reading when it is connected across a ce of e.m.f and interna resistance 0 Ω? 00 Ω Formua : Soution : i) K r 0 Ω? Ir I Current through the circuit is given by I K ii) r r.7 + [.08 ] 0.08 r.7 Ω + r Ω 00 Ω 0. Four resistances Ω, Ω, Ω and Ω form a Wheatstone s network. Find the resistance which when connected across the Ω resistance, wi baance the network. Ω esistance connected across Ω resistance to baance the network? Formua : Soution : et resistances connected across Ω be. quivaent resistance for Ω and in parae is given by p + p 0 I 60 Ir (8 + ) Ω + Current ectricity

7 .. 80 MHSH TUTOIS SCINC. Two resistances and Y in the two gaps of a meter-bridge give a nu point dividing the wire in the ratio :. If each resistance is increased by 0 Ω, the nu point divides the wire in the ratio : 6, cacuate each resistance. Y + 0 Y + 0? Y? Formua : 6 Y Y Soution : From st condition Y Y... (i) From nd condition. q u a e n g t h s o f m a g n a n i n ( ρ ) and nichrome (ρ ) are joined in the eft gap and right gap of meter-bridge. The nu point is 0 cm from the eft end. Compare the diameters of the wires. [ ρ Ωm, ρ 0 6 Ωm] engths of magnanin and nichrome in the eft and right gaps of meter bridge. 0 cm cm ρ Ωm ρ 0 6 Ωm d d where Formua :? d diameter of magnanin wire d diameter of nichrome wire + 0 Y ( + 0) (Y + 0) Y + 0 () + 80 Y + 0 (Y) + 80 Y + 0 Y + 80 Y + 0 Y 0 Ω... (ii) From equation (i) and (ii) (0) 60 0 Ω Soution : ρ ρ ρ ρ so π d and π d Current ectricity

8 MHSH TUTOIS SCINC.. 8 ρ πd ρ π d and ρ π d ρ π d ρ ρ π d πd ccording to baancing condition, d d d d d d d d 0 60 ρ d ρ d becomes, d 0 d Determine the current fowing through the gavanometer (G) as shown in the figure. Formua : Soution : et I and I be the current through B and D. I g be current through (G) G 0 Ω (Given) I g? i) ΣI 0 ii) Σ + ΣI 0 ppying Kirchhoff s nd aw to oop BD I 0 I g + I 0 I I g + I 0...(i) ppying Kirchhoff s nd aw to oop BCDB 0 (I I g ) + 0 (I + I g ) + 0 I g 0 0 I + 0 I g + 0 I + 0 I g + 0 I g 0 0I + 0 I + 0 I g 0 I + I + I g 0 Substract equations (ii) and (i) I + I I 0 I + I + I 0 + Ω Ω I I g g I 6I 0 g B G D Ig I I g 0 Ω 0 Ω 0 Ω C I +I g...(ii) I 6 I g...(iii) Current ectricity

9 .. 8 MHSH TUTOIS SCINC From equations (ii) and (iii) I + (6I g ) + I g 0 I + I g + I g 0 I 6 I g...(iv) dding equations (iii) and (iv), we have I + I 6 I g + 6 I g I + I Ig I g [ I + I ] I g I g Current ectricity. potentiometer wire has a ength of Ω and resistance of m. What resistance must be connected in series with the potentiometer wire and a ce of e.m.f having interna resistance Ω to get a potentia drop of 00 µ/mm aong the wire? m Formua : Ω Interna resistance of ce, r Ω K 00 µ/mm / /m? K I I + r+ Soution : Since I K + r+ I r K ( + + ) 0. ( + + ) 0. (6 + ) Ω. uniform wire is cut into two pieces such that one piece is twice as ong as the other. The two pieces are connected in parae in the eft gap of a meterbridge. When a resistance of 0 Ω is connected in the right gap, the neutra point is obtained at a distance of 60 cm from the right end of wire. Find the resistance of the wire before it was cut into two pieces. Formua : Soution : 0 Ω 0 cm 60 cm esistance of entire wire?. et, ength of smaer piece of wire ength of arger piece of wire Correspondingy resistance of the pieces is and. The wires are connected paraey in the eft gap of meter bridge

10 MHSH TUTOIS SCINC Ω In the eft gap, equivaent resistance is given by 0 6. With an unknown resistance in the eft gap and a resistance of 0 Ω in the right gap of meter-bridge, the nu point is obtained at 0 cm from the eft end of the wire. Find : i) the unknown resistance and ii) the shift in the position of the nu point. a) when the resistances in both the gaps are increased by Ω and b) when the resistance in each gap is shunted by a resistance of 8 Ω. 0 Ω Ω Now entire resistance of wire Ω 0 cm 60 cm i) Unknown resistance ()? ii) Formua : Soution : Shift in the position of the nu point a) when the resistances in both the gaps are increased by Ω and b) when the resistance in each gap is shunted by a resistance of 8 Ω.. i) Ω ii) a) When the resistance in both the gaps are increased by Ω Ω Ω Since cm cm Shift in nu point.7 0 Shift in nu point.7 cm towards right b) When the resistance in each gap is shunted by a resistance of 8 Ω new resistance in eft gap Current ectricity

11 .. 8 MHSH TUTOIS SCINC Ω New resistance in right gap Ω cm 7. cm shift in nu point shift in nu point 7. cm towards right 7. skeeton cube is made of wires each of resistance rω connected to a ce of e.m.f. and of negigibe interna resistance. Use Kirchhoff s aws to find the resistance between i) adjacent corners of the cube i.e. between two ends of any wire across any one edge. ii) the diagonay opposite corners of same face of cube. i.e. across face diagona. Soution : et a current x + y enter the junction of the cube BCDFGH. From the symmetry of the parae paths, current distribution wi be as shown in figure. ppying Kirchoff s second aw to the oop DHGCD, we get (y z) r + (y z) r + (y z)r zr 0 y r z r 0 z y z y ppying Kirchhoff s second aw to the oop BCD, we get xr yr zr yr 0 or x y z 0 x y y 0 z y x H y z (y z) y x + y y z D y y y; x et be the resistance across B. Then P.D. across B xr i.e., (x + y) xr 0 x x + xr x z z y 7 r B y G y z F y z C x + y Current ectricity

12 MHSH TUTOIS SCINC rω et be the resistance across C (i.e. opposite corners of the oop DCB) P.D. across C D + DC (x + y) (y + z)r y y + y 9yr y y + r 9 r 8 r Ω 8. Two cois are connected in series in one gap of meter-bridge and nu point is obtained at the midde of the wire by putting 0 Ω resistance in the other gap. The two cois are then connected in parae and it is found that the resistance in the other gap has to be decreased by 8 Ω to get the nu point at the same pace as before. Find the resistance of the cois. Soution : et and be the resistance of two cois. quivaent resistance in series s ( + ) From first condition (i) From second condition and are in parae quivaent resistance p + p + esistance in other gap 0 8 Ω Since nu point remains at mid point p Using equation (i) 0 0 Substituting in equation (i) we have ( 0) 0( 0) 0 ( 0)( 0) 0 ither 0 0 or 0 0 If 0 Ω, If 0 Ω then 0 Ω Then, 0 Ω 9. Find the radius of the wire of ength m needed to prepare a coi of resistance Ω (estivity of materia of wire is. 0 7 Ωm) m Ω ρ. 0 7 Ωm Formua : r? ρ Current ectricity

13 .. 86 MHSH TUTOIS SCINC Soution : ρ ρ π r r r r m 0.6 mm K ( + ) + K Ω 0. The resistance of a potentiometer wire is 8 Ω and its ength is 8 m. resistance box and a battery are connected in series with it. What shoud be the resistance in the box, if it is desired to have potentia drop of µ/mm? 8 Ω 8 m K µ/mm 6 0 /m 0 0 /m? Formua : K Soution : Since, I so + I + K Current ectricity becomes,

13.Current Electricity Marks :03/04

13.Current Electricity Marks :03/04 Q. State and explain kirchhoff s laws for an electric network. Ans:- Kirchhoff s law for an electric network :- In 1842 Kirchhoff s stated two laws to determine currents