To determine the relationship between the upward force W and the downward force P for the deflection to be zero at the free end of the cantilever beam, we can analyze the equilibrium conditions and apply the principles of structural mechanics.

Let's break down the problem step by step:

1. Equilibrium conditions:
- The sum of all vertical forces acting on the beam must equal zero.
- The sum of all moments about any point on the beam must equal zero.

2. Deflection at the free end:
- If the deflection at the free end is zero, it means that the beam is in static equilibrium, and the downward force P is balanced by the upward force W.

3. Equilibrium equations:
- From the equilibrium conditions, we can write the following equations:

ΣFy = 0 (sum of vertical forces)
ΣM = 0 (sum of moments)

4. Vertical forces:
- There are two vertical forces acting on the beam: the upward force W and the downward force P.
- The upward force W is applied at the mid-span of the beam.
- The downward force P is applied at the free end of the beam.
- Therefore, the sum of vertical forces can be written as:

W - P = 0

5. Moments:
- To simplify the analysis, let's consider the mid-span of the beam as the reference point for calculating moments.
- The moment generated by the upward force W at the mid-span is:

Mw = W * (L/2)

- The moment generated by the downward force P at the free end is:

Mp = P * L

- The sum of moments can be written as:

Mw - Mp = 0

6. Solving the equations:
- From equation (4), we have W - P = 0, which implies W = P.
- Substituting W = P into equation (6), we have:

Mw - Mp = 0
W * (L/2) - P * L = 0
(L/2) * W - L * P = 0

- Dividing both sides of the equation by L, we get:

(1/2) * W - P = 0

- Rearranging the equation, we obtain:

W = 2P

Therefore, the relationship between the upward force W and the downward force P for the deflection to be zero at the free end of the cantilever beam is W = 2P. Hence, option 'B' is the correct answer.