1. How much charge is represented by 4 ,600 electrons? 2. Calculate the amount of charge represented by two million protons. 3. The total charge entering a terminal is given by q=5tsin4pi tmC Calculate the current at t=0.5s 4. Determine the total charge entering a terminal between t=1s and t=2s if the current passing the terminal is i=(3t^2-t)A

1. The charge represented by 4,600 electrons can be calculated using the fundamental charge of an electron, which is approximately \(1.6 \times 10^{-19}\) coulombs. Therefore, the charge represented by 4,600 electrons is:<br /><br />\[ \text{Charge} = \text{Number of electrons} \times \text{Charge of one electron} \]<br />\[ \text{Charge} = 4600 \times 1.6 \times 10^{-19} \, \text{C} \]<br />\[ \text{Charge} = 7.36 \times 10^{-16} \, \text{C} \]<br /><br />2. The charge represented by two million protons can be calculated using the fundamental charge of a proton, which is also approximately \(1.6 \times 10^{-19}\) coulombs. Therefore, the charge represented by two million protons is:<br /><br />\[ \text{Charge} = \text{Number of protons} \times \text{Charge of one proton} \]<br />\[ \text{Charge} = 2,000,000 \times 1.6 \times 10^{-19} \, \text{C} \]<br />\[ \text{Charge} = 3.2 \times 10^{-12} \, \text{C} \]<br /><br />3. To calculate the current at \(t = 0.5\) s, we can use the formula for current, which is the rate of change of charge with respect to time. Given the formula for charge \(q = 5t \sin(4\pi t) \, \text{mC}\), we can find the current by differentiating \(q\) with respect to \(t\):<br /><br />\[ i = \frac{dq}{dt} \]<br /><br />Differentiating \(q\) with respect to \(t\):<br /><br />\[ i = \frac{d}{dt} (5t \sin(4\pi t)) \]<br /><br />Using the product rule:<br /><br />\[ i = 5 \left( \sin(4\pi t) + 4\pi t \cos(4\pi t) \right) \]<br /><br />Now, substituting \(t = 0.5\) s:<br /><br />\[ i = 5 \left( \sin(4\pi \times 0.5) + 4\pi \times 0.5 \cos(4\pi \times 0.5) \right) \]<br />\[ i = 5 \left( \sin(2\pi) + 2\pi \cos(2\pi) \right) \]<br />\[ i = 5 \left( 0 + 2\pi \times 1 \right) \]<br />\[ i = 10\pi \, \text{mA} \]<br /><br />4. To determine the total charge entering a terminal between \(t = 1\) s and \(t = 2\) s, we need to integrate the current function \(i = (3t^2 - t) \, \text{A}\) over this time interval:<br /><br />\[ Q = \int_{1}^{2} (3t^2 - t) \, dt \]<br /><br />Integrating the function:<br /><br />\[ Q = \left[ t^3 - \frac{t^2}{2} \right]_{1}^{2} \]<br />\[ Q = \left( 2^3 - \frac{2^2}{2} \right) - \left( 1^3 - \frac{1^2}{2} \right) \]<br />\[ Q = \left( 8 - 2 \right) - \left( 1 - \frac{1}{2} \right) \]<br />\[ Q = 6 - 0.5 \]<br />\[ Q = 5.5 \, \text{C} \]<br /><br />Therefore, the total charge entering the terminal between \(t = 1\) s and \(t = 2\) s is 5.5 coulombs.