if an electric current I is flowing through register are R for t seconds then find the equation of heat generated
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Answer:
By definition current is rate of flow of charge, hence
I=
dt
dQ
I=
dt
d
(at−bt
2
)
I=a−2bt.............................................................(1)
For, I=0,
0=a−2bt
2bt=a
⇒t=
2b
a
..............................................................(2)
The total heat produced in resistor R is:
H=∫
0
t
I
2
Rdt
H=∫
0
t
(a−2bt)
2
Rdt ...............................from(1)
H=∫
0
t
(a
2
−4abt+4b
2
t
2
)Rdt
H=R[a
2
t−
2
4abt
2
+
3
4b
2
t
3
]
0
t
H=R[
6
6a
2
t−12abt
2
+8b
2
t
3
]
H=R
⎣
⎢
⎢
⎡
6
6a
2
(
2b
a
)−12ab(
2b
a
)
2
+8b
2
(
2b
a
)
3
⎦
⎥
⎥
⎤
.................from (2)
H=R
⎣
⎢
⎢
⎢
⎢
⎢
⎡
6
3(
b
a
3
)−3(
b
a
3
)+(
b
a
3
)
⎦
⎥
⎥
⎥
⎥
⎥
⎤
H=R
⎣
⎢
⎢
⎢
⎡
6
2(
b
a
3
)
⎦
⎥
⎥
⎥
⎤
H=
6b
a
3
R
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