Obtain the formula for the electric field due to a long thin wire of uniform linear charge density `lambda` without using Gauss's law. [Hint.use Coulomb's law directly and evaluate the necessary integral ].

Explanation

To derive the formula for the electric field due to a long thin wire of uniform linear charge density \( \lambda \) without using Gauss's law, we will directly apply Coulomb's law and evaluate the necessary integral. Here’s a step-by-step solution:

### Step 1: Define the Problem
Consider a long thin wire of length \( L \) with a uniform linear charge density \( \lambda \) (charge per unit length). We want to find the electric field \( \mathbf{E} \) at a point \( P \) located at a perpendicular distance \( R \) from the wire.

### Step 2: Consider a Small Element of Charge
Take a small element of the wire \( dx \) at a distance \( x \) from the point \( P \). The charge \( dq \) on this small element is given by:
\[
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