The charge flowing in a conductor varies with time as `Q ="at"- (1)/(2)bt^(2)+(1)/(6)ct^(3)`, where a,b,c are positive constant. Then, the current `i = (dQ)/(dt)`

Options

A. has an initial value a
B. reaches a minimum value after time `b//c`
C. reaches a maximum value after time `b//c`
D. has a minimum value

Explanation

To solve the problem, we need to find the current \( i \) as the derivative of the charge \( Q \) with respect to time \( t \). The charge \( Q \) is given by the equation:

\[ Q = at - \frac{1}{2}bt^2 + \frac{1}{6}ct^3 \]

### Step 1: Differentiate \( Q \) with respect to \( t \)

To find the current \( i \), we need to calculate the derivative of \( Q \):

\[ i = \frac{dQ}{dt} \]

Using the power rule of differentiation, we differentiate each term in the expression for \( Q \):

1. The derivative of \( at \) is \( a \).
2. The derivative of \( -\frac{1}{2}bt^2 \) is \( -bt \) (using the power rule: \( \frac{d}{dt}(t^n) = nt^{n-1} \)).
3. The derivative of \( \frac{1}{6}ct^3 \) is \( \frac{1}{2}ct^2 \) (again using the power rule).

Putting it all together, we have:

\[ i = a - bt + \frac{1}{2}ct^2 \]

### Step 2: Evaluate the current at \( t = 0 \)

To find the initial value of the current, we substitute \( t = 0 \) into the expression for \( i \):

\[ i(0) = a - b(0) + \frac{1}{2}c(0)^2 = a \]

Thus, the initial current \( i(0) \) is \( a \).

### Step 3: Find the time when the current reaches a minimum or maximum

To find when the current reaches a minimum or maximum, we need to set the derivative of the current \( \frac{di}{dt} \) to zero. First, we differentiate the expression for \( i \):

\[ \frac{di}{dt} = -b + ct \]

Setting this equal to zero to find critical points:

\[ -b + ct = 0 \implies ct = b \implies t = \frac{b}{c} \]

### Step 4: Determine the nature of the critical point

To determine whether this critical point is a minimum or maximum, we take the second derivative of \( i \):

\[ \frac{d^2i}{dt^2} = c \]

Since \( c \) is a positive constant, \( \frac{d^2i}{dt^2} > 0 \), indicating that the critical point at \( t = \frac{b}{c} \) is a minimum.

### Summary of Results

1. The expression for current is:
\[ i = a - bt + \frac{1}{2}ct^2 \]
2. The initial current at \( t = 0 \) is \( i(0) = a \).
3. The current reaches a minimum at \( t = \frac{b}{c} \).