Problem 42 - Fundamentals of Physics

Question

A battery of potential difference \(V=12 \mathrm{~V}\) is connected to a resistive strip of resistance \(R=6.0 \Omega\). When an electron moves through the strip from one end to the other, (a) in which direction in the figure does the electron move, (b) how much work is done on the electron by the electric field in the strip, and (c) how much energy is transferred to the thermal energy of the strip by the electron?

Step-by-Step Solution

Answer

a) From negative to positive terminal; b) -1.92×10^{-18} J; c) 1.92×10^{-18} J.

Step 1: Identify the direction of electron motion

Electrons move from the negative terminal to the positive terminal of a battery due to the electric field. In a closed circuit, this is opposite to the conventional current, which is shown moving from positive to negative.

Step 2: Calculate the work done on the electron

The work done on an electron by the electric field is calculated using the equation \( W = qV \), where \( q \) is the charge of the electron \( \left( q = -1.6 \times 10^{-19} \text{ C} \right) \) and \( V = 12 \text{ V} \). Since the electron has a negative charge, \( W = (-1.6 \times 10^{-19} \text{ C})(12 \text{ V}) = -1.92 \times 10^{-18} \text{ J} \). The negative sign indicates the work is done on the electron against the electric field.

Step 3: Calculate the energy transferred to the strip

For every electron passing through the strip, the energy transferred to thermal energy is equal to the magnitude of the work done, ignoring the sign. Therefore, the energy converted to heat is \( 1.92 \times 10^{-18} \text{ J} \). This is the absolute value of the work done, as energy transfer does not consider direction.

Key Concepts

Electron Motion Direction

In an electric circuit, understanding the direction in which electrons move is key to grasping how current flows. Electrons are negatively charged particles, so their movement is driven by the electric field created by a battery or power source.

They are attracted to the positive terminal and repelled by the negative terminal, which causes them to move from the negative terminal to the positive terminal of the battery. This motion is always opposite to the direction of conventional current, which is the hypothetical flow of positive charges from positive to negative.

In essence:

• Electron motion is from negative to positive terminal.
• Conventional current moves from positive to negative terminal.

It is important to remember this distinction, as it affects how we calculate other electrical properties, such as work and energy transfer, within the circuit.

Work Done on Electron

When electrons move through a circuit, work is done on them by the electric field present throughout the conductive material. This work can be quantified using the formula \[ W = qV \]where:

• W is the work done (in joules).
• q is the charge of the electron (approximately \(-1.6 \times 10^{-19}\) coulombs).
• V is the potential difference (voltage) across the circuit or component (in this example, 12 volts).

Given that electrons have a negative charge, the work done on an electron moving through the battery and circuit is \[ W = (-1.6 \times 10^{-19} \, \text{C})(12 \, \text{V}) = -1.92 \times 10^{-18} \, \text{J} \]. The negative value indicates that work is done against the field, aligning with the electron's natural tendency to oppose movement due to its negative charge.

Energy Transfer to Thermal Energy

As electrons move through a resistive material, the energy they carry doesn't disappear but transforms into another form of energy, typically thermal energy. This happens because resistance in conductive materials impedes the flow of electrons, causing collisions that generate heat.

The energy transfer to thermal energy per electron, in this exercise, is the absolute value of the work done. Thus, despite the work done having a negative sign, the energy converted to thermal energy in the resistor is \[ 1.92 \times 10^{-18} \, \text{J}\].
Key points include:

• Thermal energy results from resistance in the circuit.
• Resistors convert electrical energy into heat energy.
• The magnitude of energy transfer is represented by the absolute value of work done.

This process is fundamental for understanding how energy is dissipated in circuits and is an essential concept of thermodynamics in electrical studies.