Electric current through a conductor varies with time as I(t)`=50sin(100pi t)`. Here I is in amperes and t in seconds. Total charge that passes any point from `t=0` to ` t= (1)/(200)s` is

Options

A. 1.2 C
B. 0.36 C
C. 0.159 C
D. 0.02 C

Answer

To find the total charge that passes through a point from \( t = 0 \) to \( t = \frac{1}{200} \) seconds when the current is given by \( I(t) = 50 \sin(100 \pi t) \), we can follow these steps:

### Step 1: Understand the relationship between current and charge
The current \( I(t) \) is defined as the rate of flow of charge \( Q \) with respect to time \( t \). Mathematically, this is expressed as:
\[ I(t) = \frac{dQ}{dt} \]
Thus, the total charge \( Q \) that passes through a point during a time interval can be found by integrating the current over that interval:
\[ Q = \int_{t_1}^{t_2} I(t) \, dt \]

### Step 2: Set up the integral
In our case, we need to integrate the current from \( t = 0 \) to \( t = \frac{1}{200} \):
\[ Q = \int_{0}^{\frac{1}{200}} 50 \sin(100 \pi t) \, dt \]

### Step 3: Compute the integral
To solve the integral, we can factor out the constant:
\[ Q = 50 \int_{0}^{\frac{1}{200}} \sin(100 \pi t) \, dt \]
Now, we will compute the integral of \( \sin(100 \pi t) \):
\[ \int \sin(100 \pi t) \, dt = -\frac{1}{100 \pi} \cos(100 \pi t) + C \]
Now, we can evaluate the definite integral:
\[ Q = 50 \left[-\frac{1}{100 \pi} \cos(100 \pi t) \right]_{0}^{\frac{1}{200}} \]

### Step 4: Evaluate the limits
Substituting the limits into the integral:
\[ Q = 50 \left[-\frac{1}{100 \pi} \left( \cos(100 \pi \cdot \frac{1}{200}) - \cos(100 \pi \cdot 0) \right)\right] \]
Calculating the cosine values:
\[ \cos(100 \pi \cdot \frac{1}{200}) = \cos(\frac{\pi}{2}) = 0 \]
\[ \cos(0) = 1 \]
Thus, we have:
\[ Q = 50 \left[-\frac{1}{100 \pi} (0 - 1)\right] = 50 \left[\frac{1}{100 \pi}\right] \]
\[ Q = \frac{50}{100 \pi} = \frac{1}{2 \pi} \]

### Step 5: Calculate the numerical value
To find the numerical value of \( Q \):
\[ Q \approx \frac{1}{2 \cdot 3.14} \approx 0.159 \, \text{Coulombs} \]

### Final Answer
The total charge that passes any point from \( t = 0 \) to \( t = \frac{1}{200} \) seconds is approximately:
\[ Q \approx 0.159 \, \text{C} \]

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