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

To find the value of the unknown resistance $R_{\mathrm{CD}}$ and its accuracy in the balanced Wheatstone bridge ABCD, we start with the condition for a balanced bridge.

The bridge has four arms connected in a loop as ABCD. The resistances of the arms are:
$R_{\mathrm{AB}}=1\text{ kΩ}\pm 2.1\%=1000\Omega \pm 2.1\%$
$R_{\mathrm{BC}}=100\Omega \pm 0.5\%$
$R_{\mathrm{CD}}$ is the unknown resistance.
$R_{\mathrm{DA}}=300\Omega \pm 0.4\%$

For a balanced Wheatstone bridge, the ratio of opposite arm resistances is equal, or the product of opposite arms is equal. Specifically, for the bridge configuration ABCD:
$R_{\mathrm{AB}}\times R_{\mathrm{CD}}=R_{\mathrm{BC}}\times R_{\mathrm{DA}}$

We can solve for the unknown resistance $R_{\mathrm{CD}}$ using:
$R_{\mathrm{CD}}=\frac{R_{\mathrm{BC}}\times R_{\mathrm{DA}}}{R_{\mathrm{AB}}}$

Substituting the nominal values:
$R_{\mathrm{CD}}=\frac{100\Omega \times 300\Omega }{1000\Omega }=30\Omega$

Next, we determine the limiting error (accuracy) of $R_{\mathrm{CD}}$. Since the relationship involves multiplication and division, the relative (percentage) limiting errors are additive:
$\frac{\Delta R_{\mathrm{CD}}}{R_{\mathrm{CD}}}=\frac{\Delta R_{\mathrm{BC}}}{R_{\mathrm{BC}}}+\frac{\Delta R_{\mathrm{DA}}}{R_{\mathrm{DA}}}+\frac{\Delta R_{\mathrm{AB}}}{R_{\mathrm{AB}}}$

Substituting the percentage tolerances given:
$\%\text{ error in }{R}_{\mathrm{CD}}=0.5\%+0.4\%+2.1\%=3.0\%$

Now, we convert the percentage limiting error back to an absolute error in ohms:
$\Delta R_{\mathrm{CD}}=30\Omega \times 3.0\%=30\times 0.03=0.9\Omega$

Therefore, the value of $R_{\mathrm{CD}}$ along with its accuracy is:
$30\text{ Ω}\pm 0.9\text{ Ω}$