The voltage across a resistance R is determined using a voltmeter connected to its ends. What relative error will be made if the readings of the voltmeter are taken as the voltage applied before it was switched on ? The current intensity in the circuit is constant.

Options

A. `(R )/(R_(0))`
B. `(R_(0))/(R )`
C. `(1)/(1+(R )/(R_(0)))`
D. `(1)/(1+(R_(0) )/(R ))`

Answer

To solve the problem, we need to determine the relative error made when the readings of the voltmeter are taken as the voltage applied before the circuit was switched on. We will follow these steps:

### Step 1: Understand the relationship between voltage, current, and resistance.
The voltage (V) across a resistor (R) is given by Ohm's Law:
\[ V = I \cdot R \]
where \( I \) is the current flowing through the circuit.

### Step 2: Define the terms.
Let:
- \( R \) = actual resistance
- \( R_0 \) = resistance when the circuit is switched off (which can be considered as the initial resistance)

### Step 3: Analyze the situation.
When the circuit is switched on, the current is constant, and the voltage across the resistance is actually \( V = I \cdot R \). However, if we take the reading of the voltmeter before the circuit is switched on, we will be measuring the voltage across the resistance \( R_0 \).

### Step 4: Calculate the relative error.
The relative error in voltage can be expressed as:
\[ \text{Relative Error} = \frac{V - V_0}{V} \]
where \( V_0 \) is the voltage reading taken when the circuit is off.

Since \( V_0 = I \cdot R_0 \) and \( V = I \cdot R \), we can substitute these into the equation:
\[ \text{Relative Error} = \frac{I \cdot R - I \cdot R_0}{I \cdot R} \]

### Step 5: Simplify the equation.
Factoring out \( I \) from the numerator:
\[ \text{Relative Error} = \frac{I(R - R_0)}{I \cdot R} \]
Cancelling \( I \) (assuming \( I \neq 0 \)):
\[ \text{Relative Error} = \frac{R - R_0}{R} \]

### Step 6: Rewrite the expression.
This can be rewritten as:
\[ \text{Relative Error} = 1 - \frac{R_0}{R} \]

### Step 7: Final expression.
Thus, the relative error can also be expressed as:
\[ \text{Relative Error} = \frac{1}{1 + \frac{R_0}{R}} \]

### Conclusion:
The correct option that matches our derived formula is:
\[ \text{Relative Error} = \frac{1}{1 + \frac{R_0}{R}} \]

### Final Answer:
The relative error made if the readings of the voltmeter are taken as the voltage applied before it was switched on is:
\[ \frac{1}{1 + \frac{R_0}{R}} \]