A student is examining a bacterium under the microscope. The E. coli bacterial cell has a mass of m = 1.90 fg (where a femtogram, fg, is 10−15g) and is swimming at a velocity of v = 2.00 μm/s , with an uncertainty in the velocity of 3.00 % . E. coli bacterial cells are around 1 μm ( 10−6 m) in length. The student is supposed to observe the bacterium and make a drawing. However, the student, having just learned about the Heisenberg uncertainty principle in physics class, complains that she cannot make the drawing. She claims that the uncertainty of the bacterium's position is greater than the microscope's viewing field, and the bacterium is thus impossible to locate.
what i did is ..
(1.90*10)*(10^-15)*(1\1000) = 1.9*10^-18
3*(2*10^-6) = 6*10^-6
Δx * (1.9*10^-18)*(6*10^-6) = (6.626*10^-34)\(4*3.14)
Δx = 4.62*10^-12
but it says wrong WHY?
-Can someone please help me get the answer?
To determine the uncertainty in position (∆x) of the E. coli bacterium, we can use the Heisenberg uncertainty principle. The principle states that the product of the uncertainty in position and the uncertainty in momentum (∆p = mv) must be greater than or equal to Planck's constant (h = 6.626 × 10^-34 J·s).
Here's the correct calculation:
Mass of E. coli bacterial cell, m = 1.90 fg = 1.90 × 10^-18 kg
Velocity of E. coli bacterial cell, v = 2.00 μm/s = 2.00 × 10^-6 m/s
Uncertainty in velocity, ∆v = 3.00% of v = 0.03 × 2.00 × 10^-6 m/s = 6.00 × 10^-8 m/s
Uncertainty in momentum, ∆p = m ∆v = (1.90 × 10^-18 kg) × (6.00 × 10^-8 m/s) = 1.14 × 10^-25 kg·m/s
According to the Heisenberg uncertainty principle, ∆x ∆p ≥ h
∆x ≥ h / ∆p = (6.626 × 10^-34 J·s) / (1.14 × 10^-25 kg·m/s)
∆x ≥ 5.80 × 10^-9 m
Therefore, the uncertainty in position (∆x) of the E. coli bacterium is approximately 5.80 nm (nanometers). This is a very small uncertainty, much smaller than the length of the bacterium itself (1 μm = 1000 nm). Hence, the student can still locate and observe the bacterium under the microscope.
To help you understand why your calculation might be incorrect, let's go through the problem and the steps to find the answer.
The Heisenberg uncertainty principle states that there is a fundamental limit to the precision with which certain pairs of physical properties of a particle can be known simultaneously. In this case, the uncertainty principle is being applied to the position and velocity of the E. coli bacterial cell in the context of microscopy.
The uncertainty in velocity (Δv) is given as 3.00% of the velocity (v), which means Δv = 0.03 * 2.00 μm/s.
To calculate the uncertainty in position (Δx), we can use the uncertainty principle equation:
Δx * Δv ≥ h/(4π)
Where h is the Planck's constant (6.626 × 10^-34 J·s) divided by 2π (to simplify the equation).
So, let's substitute the given values:
Δx * 0.03 * 2.00 μm/s ≥ (6.626 × 10^-34 J·s)/(4π)
Simplifying the equation further:
Δx * 0.06 μm/s ≥ (6.626 × 10^-34 J·s)/(4π)
Now, to solve for Δx, we can isolate it:
Δx ≥ (6.626 × 10^-34 J·s)/(4π*0.06 μm/s)
Calculating this expression:
Δx ≥ (6.626 × 10^-34 J·s)/(0.24π μm/s)
Δx ≥ 6.946 × 10^-34 J·s·s/μm·π
Expressed in scientific notation:
Δx ≥ 6.946 × 10^-34 J·s·s/(3.14 μm·s)
Simplifying the units:
Δx ≥ 2.211 × 10^-34 J·s/μm
So, the uncertainty in the position is approximately Δx ≥ 2.211 × 10^-34 μm.
It seems like your calculation involves some errors or missing conversions between units. Double-check your calculations and ensure that you've followed the correct steps outlined here.
Δx = 6.51×10^−11 m
shouldn't
3*(2*10^-6) = 6*10^-6 this read as
.03*(2*10^-6) = 6*10^-6
That moves delta postion to 4.6e-14m.