Electrostatic cleaners remove small dust particles and pollen grains from air by first ionizing them, then flowing the air between the plates of a parallel-plate capacitor, parallel to the plates, where electric forces deposit charged particles on one of the electrodes. A typical pollen grain has a mass of $5.0\times {10}^{-10}$ g, the ionizer charges it with $750$ extra electrons, and a fan moves the air at $3.0$ m/s. Ignore air resistance and gravity. What minimum electric field strength is needed to deflect the grain by $3.0$ mm before it leaves the electrodes?

Verified step by step guidance

1. Step 1: Convert the mass of the pollen grain from grams to kilograms. Since 1 gram = 10^−3 kilograms, multiply the given mass (5.0×10^−10 g) by 10^−3 to express it in kilograms.
2. Step 2: Calculate the charge on the pollen grain. The charge is determined by the number of extra electrons multiplied by the elementary charge (e = 1.6×10^−19 C). Use the formula: q = (number of electrons) × e.
3. Step 3: Determine the time the pollen grain spends between the plates. The time can be calculated using the formula: t = d / v, where d is the length of the plates (not given explicitly, but implied by the problem setup) and v is the velocity of the air (3.0 m/s).
4. Step 4: Relate the deflection distance (3.0 mm, converted to meters) to the acceleration caused by the electric field. Use the kinematic equation: y = (1/2) a t^2, where y is the deflection distance, a is the acceleration, and t is the time calculated in Step 3.
5. Step 5: Calculate the minimum electric field strength required to produce the acceleration. Use Newton's second law: F = ma, where F is the force exerted by the electric field (F = qE), m is the mass of the pollen grain, and a is the acceleration derived in Step 4. Solve for E using the formula: E = F / q.

Key Concepts

Electric Field

An electric field is a region around a charged particle 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 this context, the electric field is crucial for determining how charged particles, like pollen grains, will be influenced by the forces acting on them as they move through the capacitor.

Force on a Charged Particle

The force acting on a charged particle in an electric field can be calculated using Coulomb's law, which states that the force (F) is equal to the charge (q) multiplied by the electric field strength (E). This relationship is essential for understanding how the pollen grain, once ionized and charged, will experience a force that can deflect it within the electric field created by the capacitor plates.

Kinematics of Motion

Kinematics is the branch of mechanics that deals with the motion of objects without considering the forces that cause the motion. In this scenario, understanding the kinematic equations is important to calculate the time it takes for the pollen grain to travel a certain distance and how far it will deflect under the influence of the electric field before exiting the capacitor.