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The total charge entering a terminal is given by q=5tsin4pi tmC Calculate the current at t=0.5s

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The total charge entering a terminal is given by q=5tsin4pi tmC
Calculate the current at t=0.5s

The total charge entering a terminal is given by q=5tsin4pi tmC Calculate the current at t=0.5s

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To calculate the current, we need to find the rate of change of the charge with respect to time. The current \( I \) is given by the derivative of the charge \( q \) with respect to time \( t \).<br /><br />Given:<br />\[ q = 5t \sin(4\pi t) \]<br /><br />We need to find:<br />\[ I = \frac{dq}{dt} \]<br /><br />Using the product rule for differentiation, where \( u = 5t \) and \( v = \sin(4\pi t) \):<br /><br />\[ \frac{d}{dt}(uv) = u'v + uv' \]<br /><br />First, compute the derivatives:<br />\[ u = 5t \implies u' = 5 \]<br />\[ v = \sin(4\pi t) \implies v' = 4\pi \cos(4\pi t) \]<br /><br />Now apply the product rule:<br />\[ \frac{dq}{dt} = 5 \sin(4\pi t) + 5t \cdot 4\pi \cos(4\pi t) \]<br />\[ \frac{dq}{dt} = 5 \sin(4\pi t) + 20\pi t \cos(4\pi t) \]<br /><br />Next, evaluate this derivative at \( t = 0.5 \) s:<br />\[ I = 5 \sin(4\pi \cdot 0.5) + 20\pi \cdot 0.5 \cos(4\pi \cdot 0.5) \]<br />\[ I = 5 \sin(2\pi) + 10\pi \cos(2\pi) \]<br /><br />Since \( \sin(2\pi) = 0 \) and \( \cos(2\pi) = 1 \):<br />\[ I = 5 \cdot 0 + 10\pi \cdot 1 \]<br />\[ I = 10\pi \]<br /><br />Therefore, the current at \( t = 0.5 \) s is:<br />\[ I = 10\pi \, \text{A} \]
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