Problem 3 - College Physics

Question

If it takes \(+1.6 \times 10^{-5} \mathrm{~J}\) to move a positively charged particle between two charged parallel plates, (a) what is the charge on the particle if the plates are connected to a 6.0-V battery? (b) Was it moved from the negative to the positive plate or from the positive to the negative plate?

Step-by-Step Solution

Answer

(a) The charge is \( 2.67 \times 10^{-6} \text{ C} \). (b) It moved from the positive plate to the negative plate.

Step 1: Understand the relationship between work, charge, and potential difference

The work done in moving a charge between two points with a potential difference is given by the formula: \[ W = q \cdot V \]where \( W \) is the work done, \( q \) is the charge, and \( V \) is the potential difference (voltage). The goal is to find \( q \), the charge on the particle.

Step 2: Solve for the charge using the formula

Rearrange the formula to solve for \( q \):\[ q = \frac{W}{V} \]Substitute the given values into the formula:\[ q = \frac{1.6 \times 10^{-5} \, \text{J}}{6.0 \, \text{V}} \]Calculate \( q \).

Step 3: Perform the calculation

Continuing from Step 2:\[ q = \frac{1.6 \times 10^{-5}}{6.0} \]\[ q = 2.67 \times 10^{-6} \text{ C} \]So, the charge on the particle is \( 2.67 \times 10^{-6} \text{ C} \).

Step 4: Determine the direction of movement

Since the particle is positively charged and moves when connected to a 6.0-V battery, the work done is positive. In such cases, positive charges naturally move from higher to lower potential, meaning it moved from the positive plate to the negative plate.

Key Concepts

Charge Calculation in a Circuit

When we talk about charge calculation in the context of electric circuits, we often delve into the relationship between work done, charge, and potential difference (voltage). This relationship is crucial in understanding how electrical energy is transferred between points in a circuit. The key formula to remember here is:\[ W = q \cdot V \]where \( W \) represents the work done in joules (J), \( q \) is the charge in coulombs (C), and \( V \) is the potential difference in volts (V). To find the charge \( q \) on a particle, you can rearrange this formula to:\[ q = \frac{W}{V} \]With this formula, if you know the work done and the potential difference, you can easily calculate the charge. This is useful in many practical scenarios such as determining the charge transferred across a battery or capacitor connected in a circuit.

• Ensure all units are compatible, typically, work (J), charge (C), and voltage (V).
• Use the rearranged formula to solve for the unknown variable in circuit problems.
• Remember that a large \( V \), with a small \( W \), will lead to a small \( q \), indicating fewer charges are moved.

Understanding Potential Difference

Potential difference, often termed voltage, is a measure of the work needed to move a unit charge from one point to another. It is essential in determining how much energy is involved in the transportation of charge within an electric field or circuit. Potential difference is measured in volts (V), and it dictates the electric "pressure" or "push" that drives charges through a circuit. The potential difference between two points is calculated by measuring the difference in electric potential energy per unit charge at those points. High potential difference means more work is needed to move charge, while a low potential difference requires less work.

• Volts can be thought of as the "energy per charge unit."
• Potential difference influences how much current flows in a circuit.
• In problems with constant voltage sources, such as batteries, knowing the potential difference helps solve for other variables like charge or energy used.

Understanding potential difference is vital in both theoretical and practical aspects of studying electricity. It helps predict the behavior of charges and energy distribution in electric circuits.

Direction of the Electric Field

The direction of the electric field is an important factor to consider when analyzing circuits or electric potentials. It is defined as the direction a positive charge would naturally move in the field. In a simple circuit with parallel plates connected to a battery, the electric field moves from the positive plate to the negative plate. This movement is due to the attraction of positive charges towards the negative plate, demonstrating the natural alignment along the path of least resistance.

• Electric fields are always directed from higher to lower potential.
• The field lines indicate the path that a positive charge would naturally follow.
• In equations or sketches, these directions help visualize charge movement and potential differences.

Recognizing the direction of the electric field aids in understanding whether work is done by or against the field. This underlines why, in our exercise, the charge moves from positive to negative plate, following the electric field's path.