Starting from rest, how long does it take an electron to move 1.0 cm in a steady electric field of magnitude 100 N/C?

Verified step by step guidance

1. Step 1: Identify the forces acting on the electron. The electric field exerts a force on the electron given by \( F = qE \), where \( q \) is the charge of the electron (\( -1.6 \times 10^{-19} \, \text{C} \)) and \( E \) is the magnitude of the electric field (\( 100 \; \text{N/C} \)).
2. Step 2: Use Newton's second law \( F = ma \) to find the acceleration of the electron. Rearrange the formula to \( a = \frac{F}{m} \), where \( m \) is the mass of the electron (\( 9.11 \times 10^{-31} \; \text{kg} \)). Substitute \( F \) from Step 1 into this equation.
3. Step 3: Use the kinematic equation \( x = v_0 t + \frac{1}{2} a t^2 \) to solve for the time \( t \). Since the electron starts from rest, \( v_0 = 0 \), and the equation simplifies to \( x = \frac{1}{2} a t^2 \). Substitute \( x = 1.0 \; \text{cm} \) (convert to meters: \( 0.01 \; \text{m} \)) and \( a \) from Step 2 into this equation.
4. Step 4: Rearrange the simplified kinematic equation \( x = \frac{1}{2} a t^2 \) to solve for \( t \). The formula becomes \( t = \sqrt{\frac{2x}{a}} \). Substitute the values of \( x \) and \( a \) to find \( t \).
5. Step 5: Perform the calculation to determine the time \( t \). Ensure all units are consistent (meters, seconds, etc.) and verify the result.

Key Concepts

Electric Field

An electric field is a region around a charged particle where other charged particles experience a force. The strength of the electric field is measured in newtons per coulomb (N/C) and indicates how much force a unit charge would experience. In this question, the electric field of 100 N/C exerts a force on the electron, causing it to accelerate.

Acceleration

Acceleration is the rate of change of velocity of an object with respect to time. In the context of an electron in an electric field, it can be calculated using Newton's second law, F = ma, where F is the force acting on the electron, m is its mass, and a is the acceleration. The force on the electron is determined by the electric field strength and the charge of the electron.

Kinematic Equations

Kinematic equations describe the motion of objects under constant acceleration. They relate displacement, initial velocity, final velocity, acceleration, and time. For an electron starting from rest, the relevant equation is s = ut + (1/2)at², where s is the displacement, u is the initial velocity (zero in this case), a is the acceleration, and t is the time taken to travel the distance.