A constant current i is passed through a resi | vedclass

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(D) The rate of production of thermal energy in a resistor is given by the power $P = \frac{dU}{dt} = i^2 R$.The resistance $R$ of a conductor depends

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$d$

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(D) The rate of production of thermal energy in a resistor is given by the power $P = \frac{dU}{dt} = i^2 R$.The resistance $R$ of a conductor depends on temperature $T$ as $R = R_0(1 + \alpha \Delta T)$,where $R_0$ is the initial resistance,$\alpha$ is the temperature coefficient of resistance,and $\Delta T$ is the change in temperature.As current $i$ flows through the resistor,it heats up,so the temperature $T$ increases with time $t$. Assuming the temperature rise is proportional to the time elapsed (i.e.,$\Delta T \propto t$),we can write $R(t) = R_0(1 + \alpha' t)$,where $\alpha'$ is a constant.Substituting this into the power equation:$P(t) = i^2 R_0(1 + \alpha' t) = i^2 R_0 + (i^2 R_0 \alpha') t$.This equation is of the form $y = mx + c$,which represents a straight line with a positive slope $(i^2 R_0 \alpha')$ and a positive y-intercept $(i^2 R_0)$.Among the given plots,curve $d$ shows a linear increase starting from a non-zero value,which correctly represents the relationship.

$A$ constant current $i$ is passed through a resistor. Taking the temperature coefficient of resistance into account,indicate which of the plots shown in the figure best represents the rate of production of thermal energy in the resistor.

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