Calculating Resistance Using Ohm's Law

Ohm's Law states the relationship between voltage ($V$), current ($I$), and resistance ($R$):

$R = \frac{V}{I}$

Given:

• Voltmeter reading, $V = 10 \text{ V}$
• Ammeter reading, $I = 5 \text{ A}$

First, calculate the resistance:

$R = \frac{10 \text{ V}}{5 \text{ A}} = 2 \Omega$

Determining Measurement Errors

The least count of an instrument represents the maximum possible absolute error in a measurement. The errors are:

• Least count of voltmeter, $LC_V = 500 \text{ mV} = 0.5 \text{ V}$. So, absolute error in voltage, $\Delta V = 0.5 \text{ V}$.
• Least count of ammeter, $LC_I = 200 \text{ mA} = 0.2 \text{ A}$. So, absolute error in current, $\Delta I = 0.2 \text{ A}$.

Estimating Error in Resistance

To find the estimated error in resistance ($\Delta R$), we first calculate the relative errors in voltage and current measurements:

• Relative error in voltage: $ \frac{\Delta V}{V} = \frac{0.5 \text{ V}}{10 \text{ V}} = 0.05 $
• Relative error in current: $ \frac{\Delta I}{I} = \frac{0.2 \text{ A}}{5 \text{ A}} = 0.04 $

The total relative error in resistance ($\frac{\Delta R}{R}$) can be estimated by summing the individual relative errors (a common approximation method):

$ \frac{\Delta R}{R} \approx \frac{\Delta V}{V} + \frac{\Delta I}{I} $

$ \frac{\Delta R}{R} \approx 0.05 + 0.04 = 0.09 $

Now, calculate the absolute error in resistance ($\Delta R$):

$ \Delta R = R \times \left( \frac{\Delta R}{R} \right) $

$ \Delta R = 2 \Omega \times 0.09 = 0.18 \Omega $

Therefore, the estimated error in the resistance measurement is $0.18 \Omega$.