Chapter 6: Problem 155

Calculate the velocity of a 1.0 - \(\mu\) m electron and a potential difference used to accelerate it from rest to this velocity.

Short Answer

Expert verified

\(v = \sqrt{\frac{2 * 1.6 × 10^{-19} V J}{(9.11 × 10^{-31} kg)}}\)

Step by step solution

Identify the given values in the problem

In this exercise, we are given the following information: - 1.0 - μm electron (mass of an electron: \(m = 9.11 × 10^{-31} kg\)) - The potential difference used to accelerate the electron.

Use the formula for kinetic energy

Now, let's use the formula connecting kinetic energy, charge, and potential difference to find the kinetic energy of the electron. The formula is: \(K = q * V\) Where: - K = kinetic energy - q = charge of the electron (\(1.6 × 10^{-19} C\)) - V = potential difference (in volts)

Calculate the kinetic energy

In this step, we'll plug in the given values and calculate the kinetic energy: \(K = (1.6 × 10^{-19} C) * V\) We don't have the value of the potential difference (V) given in the problem. However, since we need to find the relation between V and the velocity, let's proceed with the formula using V itself. \(K = 1.6 × 10^{-19} V J\)

Use the formula for kinetic energy and velocity

Now, we'll use the formula that relates kinetic energy and velocity, which is: \(K = \frac{1}{2} m * v^2\) Where: - K = kinetic energy - m = mass of the electron (\(9.11 × 10^{-31} kg\)) - v = velocity of the electron

Equate the two expressions for kinetic energy

We'll set the two expressions for kinetic energy equal to each other to solve for the velocity: \(1.6 × 10^{-19} V J = \frac{1}{2} (9.11 × 10^{-31} kg) * v^2\)

Solve for the velocity

Now, we'll solve the equation for the velocity (v): \(v^2 = \frac{2 * 1.6 × 10^{-19} V J}{(9.11 × 10^{-31} kg)}\) Then take the square root of both sides: \(v = \sqrt{\frac{2 * 1.6 × 10^{-19} V J}{(9.11 × 10^{-31} kg)}}\) This equation shows the relationship between the potential difference (V) and the velocity (v) of the electron. With a specific value of V, we can calculate the corresponding velocity.

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Calculate the velocity of a 1.0 - \(\mu\) m electron and a potential difference used to accelerate it from rest to this velocity.

Short Answer

Step by step solution

Identify the given values in the problem

Use the formula for kinetic energy

Calculate the kinetic energy

Use the formula for kinetic energy and velocity

Equate the two expressions for kinetic energy

Solve for the velocity

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