$F_d = 6 \pi \mu R v_s$ </br> </br> $\mu$ is the dynamic viscosity $(8.90*10^{-4}Pa*s)$ </br> $R$ is the radius of the sphere </br> $v_s$ is the particle's velocity

We slowly accelerate the particle on a translational stage until the sphere breaks from the trap. We measure the velocity at which the sphere breaks from the trap to calculate our drag force, and equate it to the trap stiffness

$F_d = k$ </br>where $k$ is our trap force

Stoke's Drag Force Measurements

Beam Power (mW) | Escape Velocity (microns/second) | Trap Force (pN) |
---|---|---|

5.5 | 20.57 | 0.44 |

8.5 | 33.49 | 0.72 |

11.7 | 40.00 | 0.86 |

15.3 | 62.60 | 1.34 |

19.0 | 84.71 | 1.82 |

23.0 | 110.77 | 2.38 |

In [61]:

```
plot(beam_strengths, trap_strengths, 'bo', label='Trap Force')
plot(line, fit, 'r-', label='Best Fit Trap Force vs Beam Strength')
title('Trap Force vs Beam Strength')
xlabel('Beam Power (mW)')
ylabel('Trap Force (pN)')
legend()
print("Slope is " + str(p[0]))
```

In [51]:

```
mu = 8.9e-4 #Pa*S
R = 2.56e-6/2 #meters
```

- T = 0.7 seconds
- Vpp = 54V

In [52]:

```
T = .7 #seconds
V = 54 #Volts
dist = (20e-6*54)/75
v_s = dist/T
print("v_s is " + str(v_s))
F_d = 6*pi*mu*R*v_s
print("F_d is " + str(F_d))
k1 = F_d
```

- T = 0.43 seconds
- Vpp = 54V

In [53]:

```
T = .43 #seconds
V = 54 #Volts
dist = (20e-6*54)/75
v_s = dist/T
print("v_s is " + str(v_s))
F_d = 6*pi*mu*R*v_s
print("F_d is " + str(F_d))
k2 = F_d
```

- T = 0.36 seconds
- Vpp = 54V

In [54]:

```
T = .36 #seconds
V = 54 #Volts
dist = (20e-6*54)/75
v_s = dist/T
print("v_s is " + str(v_s))
F_d = 6*pi*mu*R*v_s
print("F_d is " + str(F_d))
k3 = F_d
```

- T = 0.23 seconds
- Vpp = 54V

In [55]:

```
T = .23 #seconds
V = 54 #Volts
dist = (20e-6*54)/75
v_s = dist/T
print("v_s is " + str(v_s))
F_d = 6*pi*mu*R*v_s
print("F_d is " + str(F_d))
k4 = F_d
```

- T = 0.17 seconds
- Vpp = 54V

In [56]:

```
T = .17 #seconds
V = 54 #Volts
dist = (20e-6*54)/75
v_s = dist/T
print("v_s is " + str(v_s))
F_d = 6*pi*mu*R*v_s
print("F_d is " + str(F_d))
k5 = F_d
```

- T = 0.13 seconds
- Vpp = 54V

In [57]:

```
T = .13 #seconds
V = 54 #Volts
dist = (20e-6*54)/75
v_s = dist/T
print("v_s is " + str(v_s))
F_d = 6*pi*mu*R*v_s
print("F_d is " + str(F_d))
k6 = F_d
```

In [58]:

```
beam_strengths = [5.5, 8.5, 11.7, 15.3, 19.0, 23.0] #mW
trap_strengths = array([k1, k2, k3, k4, k5, k6]) #N
trap_strengths = trap_strengths*1e12 #convert to pN
```

In [59]:

```
plot(beam_strengths,trap_strengths, 'bo')
p = polyfit(beam_strengths, trap_strengths, 1)
line = linspace(5,30,2)
fit = p[1] + p[0]*line
plot(line, fit, 'r-')
```

Out[59]:

These calculations turned out better than I had imagined they would. There is bit of uncertainty in deciding exactly when the spheres escape the optical trap, but it worked out quite well here.

Using this data, we can extrapolate what our Trap Force would be at the maximum intensity of 67.8mW.

In [60]:

```
p[1] + p[0]*67.8
```

Out[60]:

We would expect a trap force of 7.23pN at an intensity of 67.8mW