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Question
Derive the expression for the heat produced due to a current ‘I’ flowing for a time interval ‘t’ through a resistor ‘R’ having a potential difference ‘V’ across its ends. With which name is the relation known? How much heat will an instrument of 12W produce in one minute if it is connected to a battery of 12V?
Solution
Consider a resistor of resistance R. Let the current flowing through this resistor be I and the potential difference across it be V.
In time t, let Q amount of charge flows through the resistor.
Work done on moving this charge, W = VQ ... (1)
According to the definition of electric current,
`1=Q/t`
Q = I × t
Putting this in equation (1),
W = V × I × t
This work done is dissipated as heat.
Hence,
Heat produced, H = W = VIt
H = VIt ... (2)
According to Ohm’s law, V = IR.
Putting this in equation (2),
H = IR × It
H = I2Rt
This relation is known as Joule’s law of heating.
Power, P = 12 W
Potential difference, V =12 V
Time duration of current flow, t = 1 min = 60 s
`P = H/t`
H = P × t
= 12 W × 60 s
= 720 J
The heat generated by the instrument of 12 W in a minute is 720 J.
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