Problem 33 - Fundamentals of Physics

Question

A block in the shape of a rectangular solid has a cross-sectional area of \(3.50 \mathrm{~cm}^{2}\) across its width, a front-to-rear length of \(15.8 \mathrm{~cm},\) and a resistance of \(935 \Omega .\) The block's material contains \(5.33 \times 10^{22}\) conduction electrons \(/ \mathrm{m}^{3}\). A potential difference of \(35.8 \mathrm{~V}\) is maintained between its front and rear faces. (a) What is the current in the block? (b) If the current density is uniform, what is its magnitude? What are (c) the drift velocity of the conduction electrons and (d) the magnitude of the electric field in the block?

Step-by-Step Solution

Answer

(a) Current is 0.0383 A. (b) Current density is 109.4 A/m². (c) Drift velocity is 1.28 × 10⁻⁴ m/s. (d) Electric field is 226.6 V/m.

Step 1: Finding the Current in the Block

According to Ohm's Law, the current \( I \) through a conductor is given by \( I = \frac{V}{R} \), where \( V \) is the potential difference and \( R \) is the resistance. Given \( V = 35.8 \text{ V} \) and \( R = 935 \Omega \), we have:\[I = \frac{35.8}{935} = 0.0383 \text{ A}.\]Thus, the current in the block is \( 0.0383 \text{ A} \).

Step 2: Calculating the Current Density

Current density \( J \) is defined as the current \( I \) per cross-sectional area \( A \). The formula is \( J = \frac{I}{A} \). Given \( A = 3.50 \text{ cm}^2 = 3.50 \times 10^{-4} \text{ m}^2 \), and \( I = 0.0383 \text{ A} \), we have:\[J = \frac{0.0383}{3.50 \times 10^{-4}} = 109.4 \text{ A/m}^2.\]Thus, the current density is \( 109.4 \text{ A/m}^2 \).

Step 3: Determining the Drift Velocity

The drift velocity \( v_d \) can be found using the formula \( J = n e v_d \), where \( n \) is the electron density and \( e = 1.60 \times 10^{-19} \text{ C} \) is the charge of an electron:\[109.4 = (5.33 \times 10^{22})(1.60 \times 10^{-19}) v_d.\]Solving for drift velocity \( v_d \):\[v_d = \frac{109.4}{(5.33 \times 10^{22}) \cdot (1.60 \times 10^{-19})} = 1.28 \times 10^{-4} \text{ m/s}.\]

Step 4: Calculating the Electric Field in the Block

The electric field \( E \) in a conductor with a potential difference \( V \) and length \( L \) is \( E = \frac{V}{L} \). Given that \( L = 15.8 \text{ cm} = 0.158 \text{ m} \), we compute:\[E = \frac{35.8}{0.158} = 226.6 \text{ V/m}.\]Thus, the electric field's magnitude in the block is \( 226.6 \text{ V/m} \).

Key Concepts

Drift Velocity

Drift velocity is a crucial concept when dealing with electric currents in conductors. It beautifully explains how millions of tiny charge carriers, typically electrons, move through a conductor when an external electric field is applied.
In simple terms, drift velocity refers to the average speed at which these electrons "drift" in response to an electric field. Although electrons travel at incredibly high speeds due to their thermal energy, their actual net movement or drift due to an applied field is quite slow. This is because they frequently collide with atoms in the conductor.
In mathematical terms, the drift velocity ( v_{d} ) is calculated using the formula: \( J = n \, e \, v_{d} \), where:

• \( J \) is the current density, indicating how much electric current flows through a unit area.
• \( n \) is the charge carrier density, showing how many charge carriers (like electrons) there are per unit volume.
• \( e \) is the elementary charge, equal to about \( 1.60 \times 10^{-19} \, C \).

Understanding drift velocity provides insight into why current flows at the macro level despite the random motion of electrons at the micro level.

Current Density

Current density is another essential aspect of understanding electricity in conductive materials. It refers to the amount of electric current passing through a unit area of a material.
This concept is especially useful when analyzing how electricity distributes across the cross-section of conductive materials, like wires or, in our case, a block.
The formula for current density ( J ) is given as:\[ J = \frac{I}{A} \], where:

• \( I \) represents the current flowing through the conductor.
• \( A \) is the cross-sectional area through which the current passes.

Current density plays a critical role in determining how thick a wire needs to be for it to safely carry a given current without overheating or failing. Understanding this helps engineers design more efficient electrical circuits.
Different materials have varied current carrying capacities which depend on their resistivity and cross-sectional areas.

Electric Field

The electric field is a fundamental concept in the physics of electricity and electromagnetism. It describes how charged particles exert force over a distance without direct contact. This invisible field directs the movement of charges, like electrons, within a conductor.
An electric field is created in a conductor when a potential difference (voltage) is applied. This is what drives the current from a point of higher potential to a point of lower potential.
In a conductor, the magnitude of the electric field ( E ) can be calculated by the equation: \[ E = \frac{V}{L} \], where:

• \( V \) is the potential difference applied across the conductor.
• \( L \) represents the length of the conductor where the field exists.

Understanding electric fields helps explain why currents flow and the impact of potential differences across conductors. It is crucial for designs in electronics, affecting everything from small circuits to large-scale electrical grids.

Resistance in Conductors

Resistance is a vital concept to understand when studying the flow of electricity through conductors. It is a measure of how much a material opposes the direct flow of electric current.
Resistance is influenced by several factors:

• The material of the conductor: Conductors like copper have low resistance, whereas insulators like rubber have high resistance.
• The cross-sectional area: A larger area allows more room for electrons to pass through, reducing resistance.
• The length of the conductor: Longer conductors have more resistance as electrons have a longer path to travel.
• Temperature: As temperature increases, resistance usually increases in conductors due to more frequent collisions between electrons and atoms.

The formula given by Ohm's Law, \( R = \frac{V}{I} \), where \( R \) is the resistance, \( V \) is the voltage, and \( I \) is the current, helps calculate the resistance in circuits.
Understanding resistance is crucial in electronic and electrical engineering, as it impacts how efficiently devices operate and determines safe operational limits for electrical components.