A balanced Wheatstone bridge ABCD has the following arm resistances :R_AB =1 kΩ ±2.1%; R_BC =100 Ω±0.5%;=RCD is an unknown resistance; R_DA = 300 Ω±0.4%. The value of R_CD and its accuracy is

The correct option is 30 Ω ± 0.9 Ω.

Step-by-step Derivation:

1. Understanding the Bridge Balance Condition:
For a Wheatstone bridge ABCD to be balanced, the ratio of the resistances in adjacent arms must be equal. The balance condition is given by the formula:
$$R_{\mathrm{CD}}=R_{\mathrm{BC}}·\left(\frac{R_{\mathrm{DA}}}{R_{\mathrm{AB}}}\right)$$
Here, the given values of the resistances are:
$$R_{\mathrm{AB}}=1k\Omega =1000\Omega$$
$$R_{\mathrm{BC}}=100\Omega$$
$$R_{\mathrm{DA}}=300\Omega$$

Substituting these values to calculate the nominal resistance value of $R_{\mathrm{CD}}$:
$$R_{\mathrm{CD}}=100·\left(\frac{300}{1000}\right)=30\Omega$$

2. Calculating the Accuracy (Limiting Error):
When quantities are multiplied or divided, their relative limiting errors (or percentage errors) are additive. Therefore, the relative limiting error of $R_{\mathrm{CD}}$ is the sum of the relative limiting errors of the individual arms:
$$\frac{\delta R_{\mathrm{CD}}}{R_{\mathrm{CD}}}=\frac{\delta R_{\mathrm{AB}}}{R_{\mathrm{AB}}}+\frac{\delta R_{\mathrm{BC}}}{R_{\mathrm{BC}}}+\frac{\delta R_{\mathrm{DA}}}{R_{\mathrm{DA}}}$$
Substituting the given percentage errors:
$$\%Error\; in{R}_{\mathrm{CD}}=2.1\%+0.5\%+0.4\%=3.0\%$$

3. Finding the Absolute Limiting Error:
The absolute error $\delta R_{\mathrm{CD}}$ is calculated by multiplying the percentage error by the nominal value:
$$\delta R_{\mathrm{CD}}=30\Omega ·\left(\frac{3}{100}\right)=0.9\Omega$$

Thus, the value of the unknown resistance and its accuracy is:
$$R_{\mathrm{CD}}=30\Omega \pm 0.9\Omega$$