A rectangular loop of sides $$12$$ cm and $$5$$ cm, with its sides parallel to the $$x$$-axis and $$y$$-axis respectively moves with a velocity of $$5$$ cm s$$^{-1}$$ in the positive $$x$$ axis direction, in a space containing a variable magnetic field in the positive $$z$$ direction. The field has a gradient of $$10^{-3}$$ T cm$$^{-1}$$ along the negative $$x$$ direction and it is decreasing with time at the rate of $$10^{-3}$$ T s$$^{-1}$$. If the resistance of the loop is $$6$$ m$$\Omega$$, the power dissipated by the loop as heat is ______ $$\times 10^{-9}$$ W.

We have a rectangular loop of sides 12 cm and 5 cm moving with velocity $$v = 5$$ cm/s in the positive x-direction. The magnetic field is in the positive z-direction with a spatial gradient of $$\frac{dB}{dx} = -10^{-3}$$ T/cm (along the negative x-direction) and is decreasing in time at a rate $$\frac{dB}{dt} = -10^{-3}$$ T/s. The sides of the loop are parallel to the x- and y-axes, so we define $$\ell_x = 12$$ cm and $$\ell_y = 5$$ cm.

The motional EMF due to the spatial variation of the magnetic field arises because the two sides parallel to the y-axis, each of length $$\ell_y$$, experience different values of $$B$$. This EMF is given by $$\varepsilon_{\text{motional}} = \frac{dB}{dx} \cdot \ell_x \cdot v \cdot \ell_y$$ and numerically evaluates to $$\varepsilon_{\text{motional}} = 10^{-3} \times 12 \times 5 \times 5 = 300 \times 10^{-3} \text{ T cm}^2/\text{s}$$.

To express this in standard SI units, we convert $$\frac{dB}{dx} = 10^{-3} \text{ T/cm} = 0.1 \text{ T/m},\quad \ell_x = 12 \text{ cm} = 0.12 \text{ m},\quad \ell_y = 5 \text{ cm} = 0.05 \text{ m},\quad v = 5 \text{ cm/s} = 0.05 \text{ m/s},$$ so that $$\varepsilon_{\text{motional}} = 0.1 \times 0.12 \times 0.05 \times 0.05 = 3 \times 10^{-5} \text{ V}$$.

The EMF induced by the time variation of the magnetic field is $$\varepsilon_{\text{time}} = -\frac{dB}{dt}\cdot A = 10^{-3} \times 0.12 \times 0.05 = 6 \times 10^{-6} \text{ V},$$ where the negative sign indicates that $$B$$ is decreasing in time.

Since both the motional and time-varying contributions act to oppose the decrease of magnetic flux through the loop, the total induced EMF is $$\varepsilon = \varepsilon_{\text{motional}} + \varepsilon_{\text{time}} = 3 \times 10^{-5} + 6 \times 10^{-6} = 3.6 \times 10^{-5} \text{ V}$$.

The resistance of the loop is $$R = 6 \text{ m}\Omega = 6 \times 10^{-3} \,\Omega$$, so the power dissipated as heat is $$P = \frac{\varepsilon^2}{R} = \frac{(3.6 \times 10^{-5})^2}{6 \times 10^{-3}} = \frac{12.96 \times 10^{-10}}{6 \times 10^{-3}} = 2.16 \times 10^{-7} \text{ W} = 216 \times 10^{-9} \text{ W}$$.

The answer is 216.