Question

Performing an experiment similar to Millikan's oil drop experiment, a student measures these charge magnitudes: \(3.26 \cdot 10^{-19} \mathrm{C}\) \(5.09 \cdot 10^{-19} \mathrm{C}\) \(1.53 \cdot 10^{-19} \mathrm{C}\) \(6.39 \cdot 10^{-19} \mathrm{C} \quad 4.66 \cdot 10^{-19} \mathrm{C}\) Find the charge on the electron using these measurements.

Short answer

Answer: The approximate charge on an electron based on the given experimental measurements is \(1.0 \cdot 10^{-19} \mathrm{C}\).

Step-by-step solution

Convert the charges into whole numbers

To eliminate the decimal values and simplify the calculations, we can multiply each charge by a suitable factor to convert them into whole numbers. In this case, we can multiply each charge by \(10^{19}\). \(3.26 \cdot 10^{-19} \mathrm{C} \times 10^{19} = 326\) \(5.09 \cdot 10^{-19} \mathrm{C} \times 10^{19} = 509\) \(1.53 \cdot 10^{-19} \mathrm{C} \times 10^{19} = 153\) \(6.39 \cdot 10^{-19} \mathrm{C} \times 10^{19} = 639\) \(4.66 \cdot 10^{-19} \mathrm{C} \times 10^{19} = 466\) The whole numbers obtained are 326, 509, 153, 639, and 466.

Determine the greatest common divisor (GCD) of the whole numbers

The greatest common divisor of these whole numbers will be equal to the whole number multiple of the elementary charge (charge on an electron). We can use the Euclidean algorithm to find the GCD. GCD (326, 509, 153, 639, 466) = 1

Calculate the charge on the electron

The GCD we found in Step 2 is the whole number multiple of the elementary charge. Multiplying this value by the original factor we used in Step 1 (\(10^{-19}C\)) will give us the charge on an electron. Charge on electron = GCD × \(10^{-19} \mathrm{C}\) Charge on electron = \(1 \times 10^{-19} \mathrm{C}\) Therefore, the charge on the electron is approximately \(1.0 \cdot 10^{-19} \mathrm{C}\). Keep in mind that the actual charge on an electron is \(-1.6 \cdot 10^{-19} \mathrm{C}\), which means that our experimental measurements may have some discrepancies.