A conducting circular loop of area 1.0 \, mathrmm^2$1.0 \, \mathrm{m}^{2}$ is placed perpendicular to a magnetic field which varies as B = sin(100 \, t)text Tesla$B = \sin(100 \, t)\text{ Tesla}$. If the resistance of the loop is 100 \, Omega$100 \, \Omega$, then the average thermal energy dissipated in the loop in one period is ____ J.

### Related Formula phi = B cdot A$$\phi = B \cdot A$$ E = -fracdphidt$$E = -\frac{d\phi}{dt}$$ P = fracE^2R$$P = \frac{E^2}{R}$$ ### Core Logic Given area of the loop, A = 1text m^2$A = 1\text{ m}^2$ and magnetic field B = sin(100t)$B = \sin(100t)$. The magnetic flux passing through the loop is: phi = B cdot A = sin(100t) times 1 = sin(100t)$$\phi = B \cdot A = \sin(100t) \times 1 = \sin(100t)$$ Induced EMF E = left| fracdphidt right| = 100cos(100t)$E = \left| \frac{d\phi}{dt} \right| = 100\cos(100t)$ ### Step 1: Calculating Power and Energy Instantaneous power P = fracE^2R = frac100^2 cos^2(100t)100 = 100cos^2(100t)$P = \frac{E^2}{R} = \frac{100^2 \cos^2(100t)}{100} = 100\cos^2(100t)$. Thermal energy dissipated in one time period T$T$: Q = int_0^T P \, dt = int_0^T 100cos^2(100t) \, dt$$Q = \int_{0}^{T} P \, dt = \int_{0}^{T} 100\cos^2(100t) \, dt$$ The angular frequency omega = 100text rad/s$\omega = 100\text{ rad/s}$, so time period T = frac2piomega = frac2pi100 = fracpi50text sec$T = \frac{2\pi}{\omega} = \frac{2\pi}{100} = \frac{\pi}{50}\text{ sec}$. Q = 100 int_0^pi/50 cos^2(100t) \, dt = 100 int_0^pi/50 frac1 + cos(200t)2 \, dt$$Q = 100 \int_{0}^{\pi/50} \cos^2(100t) \, dt = 100 \int_{0}^{\pi/50} \frac{1 + \cos(200t)}{2} \, dt$$ Q = 50 left[ t + fracsin(200t)200 right]_0^pi/50$$Q = 50 \left[ t + \frac{\sin(200t)}{200} \right]_{0}^{\pi/50}$$ Q = 50 left( fracpi50 - 0 right) = pitext Joules$$Q = 50 \left( \frac{\pi}{50} - 0 \right) = \pi\text{ Joules}$$ ### Pattern Recognition For a sinusoidal signal, the integral of cos^2(omega t)$\cos^2(\omega t)$ over one full period T$T$ is always T/2$T/2$. Thus, int P \, dt = P_textmax times fracT2$\int P \, dt = P_{\text{max}} \times \frac{T}{2}$. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 12 Physics: Electromagnetic Induction Class 12 Physics: Alternating Current