1. Given data

• Force applied by each person: $F = 200\,\text{N}$
• Young's modulus: $Y = 1 \times 10^{11}\,\text{N m}^{-2}$
• Original length: $L = 2\,\text{m}$
• Cross-sectional area: $A = 2\,\text{cm}^2 = 2 \times 10^{-4}\,\text{m}^2$

2. Important concept

When two persons pull the wire from opposite ends with equal force $200\,\text{N}$, the tension in the wire is $200\,\text{N}$, not $400\,\text{N}$.

So, the stretching force to be used is:

$T = 200\,\text{N}$

3. Formula for extension

The extension of a wire is given by:

$\Delta L = \frac{FL}{AY}$

Substitute the values:

$\Delta L = \frac{200 \times 2}{(2 \times 10^{-4})(1 \times 10^{11})}$

4. Calculation

First compute the denominator:

$(2 \times 10^{-4})(1 \times 10^{11}) = 2 \times 10^7$

Now,

$\Delta L = \frac{400}{2 \times 10^7} = 2 \times 10^{-5}\,\text{m}$

5. Convert into micrometres

Since,

$1\,\mu\text{m} = 10^{-6}\,\text{m}$

Therefore,

$2 \times 10^{-5}\,\text{m} = 20 \times 10^{-6}\,\text{m} = 20\,\mu\text{m}$

6. Final answer

The wire extends by:

$\boxed{20\,\mu\text{m}}$