Problem 61 - University Physics with Modern Physics

Question

An electric bus operates by drawing direct current from two parallel overhead cables, at a potential difference of 600 V, and spaced 55 cm apart. When the power input to the bus's motor is at its maximum power of 65 hp, (a) what current does it draw and (b) what is the attractive force per unit length between the cables?

Step-by-Step Solution

Answer

(a) 80.82 A, (b) 1.89 x 10^-3 N/m

Step 1: Convert Horsepower to Watts

The problem states the bus motor's power in horsepower (hp). First, convert this power to watts (W). 1 hp equals 746 watts, so for a power of 65 hp, the conversion is:\[65 \text{ hp} \times 746 \text{ W/hp} = 48490 \text{ W}\]

Step 2: Calculate Current Using Power Formula

With the power (P) known and the voltage (V) given as 600 V, use the formula for electric power:\[ P = V \times I \]where I is the current. Rearrange to solve for the current, I:\[ I = \frac{P}{V} = \frac{48490 \text{ W}}{600 \text{ V}} = 80.82 \text{ A} \]

Step 3: Apply the Formula for Force Between Two Parallel Wires

The attractive force per unit length \( f \) between two parallel conductors carrying current is given by the formula:\[f = \frac{\mu_0 \cdot I^2}{2 \pi \cdot d}\]where \( \mu_0 = 4\pi \times 10^{-7} \text{ N/A}^2 \) is the permeability of free space, \( I \) is the current, and \( d \) is the distance between the wires (0.55 m). Substituting the values:\[f = \frac{4\pi \times 10^{-7} \times (80.82)^2}{2\pi \times 0.55} \approx 1.89 \times 10^{-3} \text{ N/m}\]

Key Concepts

Direct Current

Direct current (DC) is a type of electrical current that flows consistently in a single direction. Unlike alternating current (AC), which changes direction periodically, DC maintains a steady flow. This characteristic makes it particularly useful for applications where a constant voltage or current supply is needed. For instance, batteries and solar cells provide DC power.
Calculation of electric power in DC circuits is straightforward. The fundamental formula to calculate electric power is given by \( P = V \times I \), where \( P \) stands for power in watts, \( V \) is the voltage in volts, and \( I \) is the current in amperes. This formula indicates that power is the product of voltage and current, highlighting the consistent relationship between them in a DC system.

Parallel Conductors

Parallel conductors are conductors arranged so that electric current flows through them side by side. In power transmission and distribution, parallel conductors can often be found running alongside each other, carrying current to different parts of a circuit or system.
When current flows through parallel conductors, they can exert mechanical forces on each other. The magnitude of these forces depends on the current, the separation between the conductors, and the length over which the current is distributed. This interaction is a key factor in designing electrical systems, as it can affect the structural integrity and reliability of the system.

Force Between Current-Carrying Wires

When two parallel wires carry electric current, they exert a force upon each other. This force is attractive if the currents are in the same direction and repulsive if they are in opposite directions. The force per unit length \( f \) is calculated using the formula:

• \( f = \frac{\mu_0 \cdot I^2}{2\pi \cdot d} \),

where \( \mu_0 \) is the permeability of free space, \( I \) is the current through the wires, and \( d \) is the distance between them.
This formula shows how the force is directly proportional to the square of the current and inversely proportional to the distance between the wires. This principle is essential in the field of electromagnetism, as it helps to understand how electric circuits are affected by magnetic fields created by currents in other nearby circuits.