How do we know that the charge of one electron is 1.602X10^-16?

The charge of one electron, approximately 1.602×10−191.602 \times 10^{-19}1.602×10−19 coulombs (not 10−1610^{-16}10−16), was measured through the famous Millikan oil drop experiment in 1909, conducted by American physicist Robert A. Millikan. Here’s how Millikan’s experiment helped in determining the charge of an electron:

Millikan Oil Drop Experiment

1. Setup: Millikan sprayed tiny oil droplets into a chamber with an electric field created between two charged plates. The oil droplets would pick up extra electrons from friction or radiation, giving them a small negative charge.

2. Balancing Forces: By adjusting the voltage of the electric field, Millikan could make some of the oil droplets suspend in mid-air, meaning the upward electric force was exactly balancing the downward gravitational force on each droplet. At this point:

   qE=mgqE = mgqE=mg

   where qqq is the charge on the droplet, EEE is the electric field, mmm is the mass of the droplet, and ggg is the gravitational acceleration.

3. Calculating the Charge: Millikan measured the droplet’s mass by observing its fall rate in the absence of an electric field. By knowing mmm, ggg, and EEE, he calculated qqq, the total charge on each droplet.

4. Quantization of Charge: Millikan repeated the experiment for many droplets and noticed that the charges he measured were always multiples of a smallest, constant value: 1.602×10−191.602 \times 10^{-19}1.602×10−19 coulombs. He concluded that this smallest value was the fundamental charge, the charge of a single electron.

Through this precise experiment, Millikan provided the first accurate measurement of the elementary charge eee, establishing that all charges are integer multiples of this fundamental charge.

So why do we say 1C is 6.24X10^24 electrons?

The statement that "1 coulomb is 6.24×10186.24 \times 10^{18}6.24×1018 electrons" is derived from the known charge of a single electron.

Since the charge of one electron is approximately 1.602×10−191.602 \times 10^{-19}1.602×10−19 coulombs, we can calculate how many electrons make up one coulomb by dividing 1 coulomb by this fundamental charge:

1 C1.602×10−19 C/electron≈6.24×1018 electrons\frac{1 \text{ C}}{1.602 \times 10^{-19} \text{ C/electron}} \approx 6.24 \times 10^{18} \text{ electrons}1.602×10−19 C/electron1 C​≈6.24×1018 electrons

Thus, 1 coulomb of charge is equivalent to the charge carried by approximately 6.24×10186.24 \times 10^{18}6.24×1018 electrons. This large number arises because the coulomb is a macroscopic unit, while the electron's charge is very small on a microscopic scale.

How did we quantify 1C before the charge of 1 electron was discovered? In other words, how is 1C = 1J/1s?

The definition of the coulomb (1 C) evolved from practical measurements of electric current, long before the charge of a single electron was known. Here’s how the concept of 1 coulomb and its relation to current, joules, and seconds developed:

Historical Definition of 1 Coulomb

1. Early Electrical Measurements: Before understanding the atomic nature of charge, scientists could measure electric current (the flow of electric charge) through practical observations of its effects, such as heating a wire, deflecting a compass needle, or producing a magnetic field around a conductor. Current was measured by the ampere (A), which was defined based on the magnetic force produced by electric currents in conductors.

2. Definition of the Ampere: In 1948, the ampere was formally defined in terms of the force between two parallel conductors:

   • 1 ampere is the current that, when flowing through two parallel conductors placed 1 meter apart in a vacuum, produces a force of 2×10−72 \times 10^{-7}2×10−7 newtons per meter of conductor length.

3. Coulomb Derived from the Ampere: Since current is defined as the rate of charge flow, 1 coulomb was defined as the amount of charge that flows when a steady current of 1 ampere runs for 1 second:

   1 C=1 A×1 s1 \text{ C} = 1 \text{ A} \times 1 \text{ s}1 C=1 A×1 s

   This definition did not require knowledge of individual electrons; it was based on observable macroscopic quantities.

4. Relating Coulombs to Energy (Joules): The relation between the coulomb and the joule per second arises from electric potential (voltage). When 1 coulomb of charge moves across a potential difference of 1 volt, it does 1 joule of work:

   Energy (J)=Charge (C)×Potential (V)\text{Energy (J)} = \text{Charge (C)} \times \text{Potential (V)}Energy (J)=Charge (C)×Potential (V)

   This means that if 1 coulomb of charge flows through an electric circuit every second under a 1-volt potential, it delivers 1 watt of power (1 joule per second).

Summary

So, the coulomb was quantified based on electric current (ampere) and time (second) before the discovery of the electron. The energy relationship (1 C=1 J/1 s)(1 , \text{C} = 1 , \text{J} / 1 , \text{s})(1C=1J/1s) comes from this practical approach to defining electric charge and current in a way that reflects measurable physical phenomena, even before understanding the underlying particle physics.

Does the conductor exert force on each other? And why was the number 2X10-7 chosen?

Yes, in the definition of the ampere, the two parallel conductors do indeed exert a force on each other. This force arises from the magnetic fields generated by the electric currents flowing through each conductor. Here’s how it works and why that particular value, 2×10−72 \times 10^{-7}2×10−7 newtons per meter, was chosen:

Magnetic Force Between Parallel Currents

1. Magnetic Field Due to Current: When an electric current flows through a conductor, it creates a magnetic field around it. For two parallel conductors with currents flowing in the same direction, the magnetic fields interact in a way that attracts the conductors toward each other. If the currents flow in opposite directions, the conductors repel each other.

2. Magnetic Force Per Unit Length: The force between two long, parallel currents I1I_1I1​ and I2I_2I2​ separated by a distance ddd is given by:

   F=μ02π⋅I1I2d per meter of conductor lengthF = \frac{\mu_0}{2 \pi} \cdot \frac{I_1 I_2}{d} , \text{per meter of conductor length}F=2πμ0​​⋅dI1​I2​​per meter of conductor length

   where μ0\mu_0μ0​ is the magnetic constant (or permeability of free space) and has a value of approximately 4π×10−7 N/A24 \pi \times 10^{-7} , \text{N/A}^24π×10−7N/A2.

3. Defining the Ampere Using Force: To make the ampere a practical unit of current, the force between two conductors carrying 1 ampere each, placed 1 meter apart, was set to exactly 2×10−72 \times 10^{-7}2×10−7 newtons per meter. This choice was made to simplify calculations and establish a standard unit for electrical current, aligning with the magnetic constant.

Why 2×10−72 \times 10^{-7}2×10−7 Newtons?

The value 2×10−72 \times 10^{-7}2×10−7 newtons per meter was chosen to make the ampere consistent with other physical constants in the International System of Units (SI). Specifically:

• Consistency with μ0\mu_0μ0​: Setting the force between two 1-ampere currents to 2×10−72 \times 10^{-7}2×10−7 newtons per meter made μ0\mu_0μ0​ exactly equal to 4π×10−7 N/A24 \pi \times 10^{-7} , \text{N/A}^24π×10−7N/A2, a value that simplifies equations in electromagnetism (particularly in the Biot–Savart law and Ampère's law).

• Practical Measurability: This definition also allowed current (the ampere) to be defined in terms of a measurable force, which was practical for experimental and engineering purposes in the early days of electromagnetism, before the discovery of the electron and quantization of charge.

Summary

So, the conductors exert a measurable magnetic force on each other due to their currents, and the value 2×10−72 \times 10^{-7}2×10−7 was chosen to set a practical, measurable standard for current, creating a straightforward relationship between the ampere, the magnetic constant μ0\mu_0μ0​, and other SI units. This simplified and unified the system of electromagnetic measurements long before we had a complete understanding of atomic charge.