An electron is released from rest at the center of a parallel-plate capacitor that has a 1.0 mm spacing. The electron then strikes one of the plates with a speed of 1.5×10⁶ m/s. What is the electric field strength inside the capacitor?

Verified step by step guidance

1. Step 1: Identify the known quantities in the problem. The distance between the plates (d) is 1.0 mm or 0.001 m, the final speed of the electron (v) is 1.5×10^6 m/s, and the initial speed (u) is 0 m/s since the electron is released from rest. The mass of the electron (m) is approximately 9.11×10^-31 kg, and the charge of the electron (q) is -1.6×10^-19 C.
2. Step 2: Use the kinematic equation to find the acceleration of the electron. The equation is: $v^2=u^2+2ad$, where v is the final velocity, u is the initial velocity, a is the acceleration, and d is the distance. Rearrange the equation to solve for acceleration (a): $a=v^2-u^2/2d$.
3. Step 3: Relate the acceleration of the electron to the electric field strength (E) using Newton's second law. The force on the electron due to the electric field is given by $F=qE$, and Newton's second law states $F=ma$. Combine these equations to express the electric field strength as $E=ma/q$.
4. Step 4: Substitute the expression for acceleration (a) from Step 2 into the equation for electric field strength (E) from Step 3. This gives: $E=m(v^2-u^2/2d)/q$.
5. Step 5: Plug in the known values for m, v, u, d, and q into the equation derived in Step 4 to calculate the electric field strength. Ensure all units are consistent (e.g., meters for distance, kilograms for mass, etc.) before performing the calculation.

Key Concepts

Electric Field

An electric field is a region around charged particles where other charged particles experience a force. It is defined as the force per unit charge and is measured in volts per meter (V/m). In a parallel-plate capacitor, the electric field is uniform between the plates and is directed from the positive plate to the negative plate.

Kinematics of Charged Particles

The motion of charged particles, such as electrons, in an electric field can be analyzed using kinematic equations. When an electron is released from rest, it accelerates due to the electric field, and its final velocity can be calculated using the equations of motion, which relate acceleration, initial velocity, final velocity, and distance traveled.

Work-Energy Principle

The work-energy principle states that the work done on an object is equal to the change in its kinetic energy. In the context of an electron in an electric field, the work done by the electric field on the electron as it moves between the plates results in an increase in its kinetic energy, which can be used to determine the electric field strength.