What potential difference is needed to accelerate an electron from rest to a speed of 2.0×10⁶ m/s?

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Identify the relationship between the kinetic energy gained by the electron and the work done by the electric field. The work done is equal to the change in kinetic energy: \( W = \Delta KE \).

Express the kinetic energy of the electron using the formula \( KE = \frac{1}{2}mv^2 \), where \( m \) is the mass of the electron (\( 9.11 \times 10^{-31} \; \text{kg} \)) and \( v \) is the final speed of the electron (\( 2.0 \times 10^6 \; \text{m/s} \)).

Relate the work done to the potential difference \( V \) using the equation \( W = qV \), where \( q \) is the charge of the electron (\( 1.6 \times 10^{-19} \; \text{C} \)).

Set the work done equal to the change in kinetic energy: \( qV = \frac{1}{2}mv^2 \). Solve for \( V \) by rearranging the equation: \( V = \frac{\frac{1}{2}mv^2}{q} \).

Substitute the known values for \( m \), \( v \), and \( q \) into the equation to calculate the potential difference \( V \).

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주요 개념

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

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion, calculated using the formula KE = 1/2 mv², where m is mass and v is velocity. For an electron accelerated from rest, its kinetic energy at a speed of 2.0×10⁶ m/s can be determined, which is essential for understanding the energy required for acceleration.

Electric Potential Difference

Electric potential difference, or voltage, is the work done per unit charge to move a charge between two points in an electric field. It is directly related to the kinetic energy gained by a charged particle, such as an electron, when accelerated through a potential difference, allowing us to calculate the necessary voltage to achieve a specific speed.

Charge of an Electron

The charge of an electron is a fundamental physical constant, approximately -1.6 × 10⁻¹⁹ coulombs. This negative charge is crucial in calculations involving electric fields and potential differences, as it determines the direction of force and energy transfer when an electron is accelerated by an electric field.