An atomic force microscope cantilever, of rectangular cross section, has a length, l, of 200μm, a width, b, of 20μm and a depth, h, of 1μm. It is made of silicon nitride, which has a Young's modulus, E, of 300GPa. What is the maximum force that can be applied with the cantilever, if the maximum strain in the beam is to be limited to 0.1%?
Pmax (in μN):
To calculate the maximum force that can be applied to the cantilever, we need to find the maximum stress the cantilever can withstand without exceeding the maximum strain limit.
The maximum strain, εmax, is given as 0.1% or 0.001.
The stress, σ, can be calculated using Hooke's law:
σ = E * ε
Where:
- σ is the stress,
- E is the Young's modulus,
- ε is the strain.
Given values:
E = 300 GPa = 300 × 10^9 Pa (Pascals)
εmax = 0.001
Substituting these values into the equation, we can calculate the maximum stress:
σmax = E * εmax
σmax = 300 × 10^9 Pa * 0.001
σmax = 300 × 10^6 Pa
Now, to calculate the maximum force, Fmax, that can be applied, we can use the formula for stress:
σ = F / A
Where:
- σ is the stress,
- F is the force,
- A is the cross-sectional area of the cantilever.
The cross-sectional area, A, of the cantilever can be calculated as:
A = l * b
Given values:
l = 200 μm = 200 × 10^(-6) m
b = 20 μm = 20 × 10^(-6) m
Substituting these values into the equation, we can calculate the cross-sectional area:
A = (200 × 10^(-6) m) * (20 × 10^(-6) m)
A = 4 × 10^(-8) m²
Now we can rearrange the stress formula to solve for the force:
F = σ * A
Substituting values, we have:
Fmax = (300 × 10^6 Pa) * (4 × 10^(-8) m²)
Fmax = 12 μN
Therefore, the maximum force that can be applied with the cantilever is 12 μN.