Problem 65 - University Physics with Modern Physics

Question

A conducting bar with mass m and length \(L\) slides over horizontal rails that are connected to a voltage source. The voltage source maintains a constant current \(I\) in the rails and bar, and a constant, uniform, vertical magnetic field \(\overrightarrow{B}\) fills the region between the rails (\(\textbf{Fig. P27.65}\)). (a) Find the magnitude and direction of the net force on the conducting bar. Ignore friction, air resistance, and electrical resistance. (b) If the bar has mass \(m\), find the distance \(d\) that the bar must move along the rails from rest to attain speed \(v\). (c) It has been suggested that rail guns based on this principle could accelerate payloads into earth orbit or beyond. Find the distance the bar must travel along the rails if it is to reach the escape speed for the earth (11.2 km/s). Let \(B =\) 0.80 T, \(I =\) 2.0 \(\times\) 10\(^3\) A, \(m =\) 25 kg, and \(L =\) 50 cm. For simplicity assume the net force on the object is equal to the magnetic force, as in parts (a) and (b), even though gravity plays an important role in an actual launch in space.

Step-by-Step Solution

Answer

(a) 800 N to the right; (b) \(d = \frac{v^2}{64}\); (c) 1,960 km.

Step 1: Calculate the Magnetic Force on the Bar

The magnetic force on the bar can be calculated using the formula for the force on a current-carrying conductor in a magnetic field: \[ F = I \cdot L \cdot B \]where:- \(I\) is the current (2.0 \(\times\) 10\(^3\) A),- \(L\) is the length of the bar (0.50 m),- \(B\) is the magnetic field (0.80 T).Substituting these values in,\[ F = (2.0 \times 10^3 \text{ A}) \cdot (0.50 \text{ m}) \cdot (0.80 \text{ T}) = 800 \text{ N} \]The direction of this force, according to the right-hand rule, will be perpendicular to both the current and the magnetic field, hence horizontally along the direction of motion.

Step 2: Find the Distance for the Bar to Reach Velocity \(v\)

We need to find the distance \(d\) that the bar must travel to attain a speed \(v\), starting from rest, under the magnetic force. The work-energy principle states that the work done by the force equals the change in kinetic energy:\[ F \cdot d = \frac{1}{2} m v^2 \]Solving for \(d\):\[ d = \frac{\frac{1}{2} m v^2}{F} \]Substituting \(F = 800 \text{ N}\) and \(m = 25 \text{ kg}\), we get:\[ d = \frac{\frac{1}{2} \times 25 \times v^2}{800} = \frac{12.5 v^2}{800} = \frac{v^2}{64} \]

Step 3: Calculate the Distance to Reach Escape Speed

The escape speed for Earth is 11.2 km/s, which is 11,200 m/s. Using the formula derived in Step 2:\[ d = \frac{(11200)^2}{64} = \frac{125440000}{64} = 1,960,000 \text{ m} \]Thus, the bar must travel a distance of 1,960,000 meters to reach the escape speed of Earth.

Key Concepts

current-carrying conductor

A current-carrying conductor is a medium through which electric current flows. Imagine a wire or a metal bar connected to a power source like a battery; that's your conductor. In our example, the conducting bar serves as this medium. The conductor experiences a magnetic force when placed in a magnetic field, which is crucial in many applications such as electric motors and rail guns.
Electric current refers to the flow of electric charges, typically measured in Amperes (A). In a practical sense, when you plug an electric appliance into an outlet, it draws current from your house's electrical wiring, which acts as the conductor.
In our case, the problem specifies a current of 2.0 \( \times \) 10³ A flowing through the conducting bar, which in the presence of the magnetic field, produces a force that pushes the bar along the rails.
Here's how it works:

• Current moves through the conductor, creating an electric field.
• When the conductor interacts with a magnetic field, it experiences a magnetic force.
• This force is perpendicular to both the direction of the current and the magnetic field, following the right-hand rule.

Considering no other resistive forces like friction or air resistance, the only significant force acting on the bar is this magnetic force. This forces causes the movement of the bar along the rail.

magnetic field

A magnetic field is an invisible force field around magnetic materials, like a magnet, that affects other materials within its range. It is represented by the symbol \( \overrightarrow{B} \) and is measured in Tesla (T).
In our exercise, a magnetic field of 0.80 T uniformly covers the area between the rails where the conducting bar is situated. This field interacts with the current in the bar, resulting in the magnetic force that propels the bar.
Understanding a magnetic field involves visualizing field lines that emanate from a magnet. These lines show the direction of the magnetic force and its influence on surrounding objects. Here's a quick breakdown:

• The direction of the field lines goes from the north to the south pole of a magnet.
• These lines are closer together where the field is stronger, indicating a higher magnetic force.
• The interaction of a magnetic field with electric current results in a force perpendicular to the direction of both the field and the current, known as the Lorentz force.

In physics applications, the combination of a magnetic field and current is often used to create motion or generate force, much like the effect in our exercise.

work-energy principle

The work-energy principle relates the work done by forces on an object to its change in kinetic energy. This fundamental physics concept helps us understand how to calculate the motion of objects, such as the conducting bar in the exercise.
According to the principle, the work done on an object is equal to its change in kinetic energy. Kinetic energy is the energy a body possesses due to its motion, given by the formula \( KE = \frac{1}{2} mv^2 \) where \(m\) is mass and \(v\) is velocity. In our scenario, we consider that:

• The magnetic force is doing work on the bar as it moves along the rails.
• This work is calculated as the product of the magnetic force and the distance over which it moves \( F \cdot d \).
• As stated by the work-energy principle: \( F \cdot d = \frac{1}{2} m v^2 \).

To find the distance \(d\) that the bar must travel to reach a certain velocity \(v\), we rearrange the formula to \( d = \frac{\frac{1}{2} m v^2}{F} \). This calculation shows how the energy used to do work can change the velocity of the bar. For example, to reach Earth’s escape velocity, the bar has to travel a vast distance due to the immense force needed.