What is the speed of an electron that has been accelerated from rest through a potential difference of 1000 V?

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Start by understanding the relationship between the potential difference and the kinetic energy gained by the electron. The work done on the electron by the electric field is equal to the change in its kinetic energy. This can be expressed as:

qV = \(\frac{1}{2}\)mv^2

, where

q

is the charge of the electron,

V

is the potential difference,

m

is the mass of the electron, and

v

is the speed of the electron.

Rearrange the equation to solve for the speed

v

v = \(\sqrt{\frac{2qV}{m}\)}

Substitute the known values into the equation. The charge of an electron is

q = 1.6 \(\times\) 10^{-19} \(\text{ C}\)

, the mass of an electron is

m = 9.11 \(\times\) 10^{-31} \(\text{ kg}\)

, and the potential difference is

V = 1000 \(\text{ V}\)

Plug these values into the formula:

v = \(\sqrt{\frac{2(1.6 \times 10^{-19}\))(1000)}{9.11 \(\times\) 10^{-31}}}

Simplify the expression under the square root to calculate the speed of the electron. Ensure that the units are consistent and verify the result for correctness.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Electric Potential Energy

Electric potential energy is the energy a charged particle possesses due to its position in an electric field. When an electron is accelerated through a potential difference, it gains kinetic energy equal to the work done on it by the electric field, which can be calculated using the formula: ΔU = qV, where q is the charge of the electron and V is the potential difference.

Kinetic Energy

Kinetic energy is the energy of an object due to its motion, expressed mathematically as KE = 1/2 mv², where m is the mass and v is the velocity of the object. In the context of an electron accelerated through a potential difference, the kinetic energy gained by the electron can be equated to the electric potential energy it acquires, allowing us to solve for its speed.

Charge of an Electron

The charge of an electron is a fundamental physical constant, approximately -1.6 x 10^-19 coulombs. This negative charge is crucial in calculations involving electric fields and potential differences, as it determines the direction of the force acting on the electron when it is subjected to an electric field, ultimately influencing its acceleration and resulting speed.