560
12 Electric Circuits
12.20 H = P 1 t 1 =
V 2
R 1
t 1
where H is joule heat and t is time
∴ t 1 =
H
V 2 R 1 (coil 1)
t 2 =
H
V 2 R 2
(coil 2)
(a) t =
H
V 2 R =
H
V 2 (R 1 + R 2 ) = t 1 + t 2 (coils in series)
(b)
1
t 1
=
V 2
H R 1
(coil 1)
1
t 2
=
V 2
H R 2
(coil 2)
∴
1
t 1
+
1
t 2
=
V 2
H
1
R 1
+
1
R 2
=
V 2
H R
=
1
t
or
t =
t 1 t 2
t 1 + t 2
(coils in parallel)
12.21 (a) i =
ξ
R 1 + R 2 + r
=
24
4 + 6 + 2
= 2 A
P 1 = i
2 R 1 = 2
2
× 4 = 16 W
P 2 = i
2 R 2 = 2
2
× 6 = 24 W
(b) Effective resistance of R 1 and R 2 in parallel is
R =
R 1 R 2
R 1 + R 2
=
4 × 6
4 + 6
= 2.4
i =
ξ
R + r
=
24
2.4 + 2
= 5.45 A
i 1 =
i R 2
R 1 + R 2
=
5.45 × 6
4 + 6
= 3.27 A
i 2 =
i R 1
R 1 + R 2
=
5.45 × 4
4 + 6
= 2.18 A
P 1 = i
2
1 R 1 = (3.27)
2
× 4 = 42.8 W
P 2 = i
2
2 R 2 = (2.18)
2
× 6 = 28.5 W