A uniform conductor of resistance R is cut into 20 equal pieces. Half of them are joined in series and the remaining half of them are connected in parallel. If the two combinations are joined in series, then the effective resistance of all the pieces is

Options

A. `R `
B. `(R )/(2 ) `
C. ` (101 R) /( 200 )`
D. ` (201 R)/(200)`

Answer

To solve the problem, we will follow these steps systematically:

### Step 1: Determine the resistance of each piece
Given that a uniform conductor of resistance \( R \) is cut into 20 equal pieces, the resistance of each piece can be calculated as:
\[ R_{\text{piece}} = \frac{R}{20} \]

### Step 2: Calculate the equivalent resistance of the series combination
Half of the pieces (10 pieces) are connected in series. The total resistance for resistors in series is the sum of their resistances. Therefore, the equivalent resistance for the series combination is:
\[ R_{\text{series}} = R_{\text{piece}} \times 10 = \frac{R}{20} \times 10 = \frac{R}{2} \]

### Step 3: Calculate the equivalent resistance of the parallel combination
The remaining half (10 pieces) are connected in parallel. The formula for the equivalent resistance \( R_{\text{parallel}} \) of \( n \) identical resistors \( R \) connected in parallel is:
\[ \frac{1}{R_{\text{parallel}}} = \frac{n}{R} \]
For our case:
\[ \frac{1}{R_{\text{parallel}}} = \frac{10}{\frac{R}{20}} = \frac{10 \times 20}{R} = \frac{200}{R} \]
Thus, the equivalent resistance for the parallel combination is:
\[ R_{\text{parallel}} = \frac{R}{200} \]

### Step 4: Combine the two equivalent resistances in series
Now, we have the equivalent resistance from the series combination \( R_{\text{series}} \) and the equivalent resistance from the parallel combination \( R_{\text{parallel}} \). Since these two combinations are connected in series, the total equivalent resistance \( R_{\text{total}} \) is given by:
\[ R_{\text{total}} = R_{\text{series}} + R_{\text{parallel}} = \frac{R}{2} + \frac{R}{200} \]

### Step 5: Simplify the total equivalent resistance
To combine these fractions, we need a common denominator. The least common multiple of 2 and 200 is 200:
\[ R_{\text{total}} = \frac{100R}{200} + \frac{R}{200} = \frac{100R + R}{200} = \frac{101R}{200} \]

### Final Answer
The effective resistance of all the pieces is:
\[ \boxed{\frac{101R}{200}} \]