HW5 Solution – PHYS 498 Spring 2012

1. Magnetic tweezers

a) From the lecture, we have derived this equation:

� � ��� ∙ �⟨�⟩

where F is magnetic force, kB is the Boltzmann constant, T is the temperature, l is the length of DNA

and δx is the distance of the bead from the center. All the variables are straight-forward to understand,

except for <δx2>. <δx2> is the average of all the distance from the bead center squared. It is inaccurate to

say that <δx2> = <δx>2.

<δx2> is the variance of the bead distribution, which follows a normal distribution. Since variance = σ2,

where σ is the standard deviation, we will be able to find <δx2> if we can find σ (<δx2> = σ2). Once σ is

known, F can be calculated easily. However, it is difficult to find σ of the bead distribution from the

diagram. What we can measure easily is the edge of the bead distribution (Rmax). Since σ is proportional

to Rmax, we can rewrite the equation as:

� � ��� ∙ � � � � ∙ 1�����

Using values of F and Rmax of the other three bead distributions (F = 3.36 pN, 0.097 pN, 0.040 pN and

Rmax = 0.25 µm, 1.12 µm, 1.55 µm), we can plot F vs 1/Rmax2 and obtain a linear relation with a slope of

0.2115 N µm2. This allows us to calculate F for Rmax of 1.12, which is 0.51 pN. This is close to the real

experimental value of 0.35 pN

b) The graph is a collection of the bead positions in horizontal (Y) and vertical (Z) positions. To get the

position in Y, we simply measure the position of the bead along the horizontal axis. To get the position

in Z, we look at the diffraction patterns produced by the bead. By first calibrating the optical trap with

a set of known Z distances, we will later be able to extract Z position information of any given

diffraction pattern.

c) The uncertainty in x depends on F as described in the following equation:

⟨�⟩ � ��� ∙ ��

At small F, the effective stiffness is small and therefore δx is big. For big F, the effective stiffness is large,

and δx is small.

d) According to equipartition theorem, at equilibrium the energy of oscillation in z direction is ½kBT. At

equlibrium, F = kl, and ΔU = Fδz = ½kBT. Therefore we have:

� � ���2�

From this equation, we can see that as F increases δz decreases.

2. Vertical (z-direction) laser trapping

a) As shown in the diagram below, the initial beam Pi is refracted inside the bead, and emerges as Pf. The bead

causes a change in momentum of the incident photon, which means that the bead is exerting a force on the

photon. The changes in momentum of the left and right rays are indicated as ΔP1 and ΔP2 respectively. These

changes in momentum allow us to figure out the resultant force exerted by the bead to the photons, which is

FB->P. FB->P is pointing upwards. The force exerted by the photons to the bead is simply the opposite, and it is

pointing downwards, as shown by FP->B. All in all, the bead will be pushed downwards towards the focus of

the trap

b) In the absence of scattering or absorption, the bead would be trapped such that the rays are focused at the

center of the bead.

c) The bead will be trapped below the focus. The scattered light exerts a force that pushes the bead down. The

force to balance this downward push from scattered light would occur at a point below the focus

3. Optical Traps

a) Using the values of Q = 0.1, n = 1.5, P = 100 mW, c = 3 x 108 m/s and λ = 1 µm, we find:

� � ������ � 7.5 ∙ 10�� ! � 0.075 # �!

b) Using the values of kB = 1.38 x 10-23 J/K, T = 300 K and k = 7.5 x 10-5 N/m, we find: 12�� � 12���

� $���� � 7.4�! & 5�!

The step size of myosin is slightly smaller than the thermal noise. Using appropriate analysis (reducing the

bandwidth of data collected), it is possible to measure the step size of myosin. In fact, we will be able to get

Angstrom level resolution by reducing the bandwidth

c)

� � � � 0.075 # �! ∙ 7.4�! � 0.56# ( 4#

Since the minimum force the trap can measure is smaller than the 4 pN force exerted by myosin, the optical

trap should be able to easily measure the power-stroke force

4. Odd shaped bead

As shown in the ray diagram above, the gradient of light exerts a rightward force on the bead. This shows that

the bead is moving towards a region of higher light intensity

5. Optical trap, another view

a) The dipole moment (p) is related to the electric field (E) through the equation:

p = α E

where α is the polarizability of the material

b) ) � *+ ∙ , � *- ∙ |,|�

� � *) � - ∙ |,|� � 2-,/,/

As E increases, the force increases because it is proportional to E. As dE/dx increases, the force will also

increase

If we imagine a Gaussian beam that is most intense in the center, we find that dE/dx > 0 to the left of the peak,

which produces a positive force that directs the bead to the right. When the bead is to the right of the peak, the

force will be negative and it will direct the bead to the left. This way the bead will be moved towards region of

higher laser intensity.

c) Intensity is proportional to |E|2

0 � - ∙ |,|� � - ∙ 1