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Chapter 22: Problem 31

ssm Two 0.68 -m-long conducting rods are rotating at the same speed in opposite directions, and both are perpendicular to a \(4.7-\mathrm{T}\) magnetic field. As the drawing shows, the ends of these rods come to within 1.0 \(\mathrm{mm}\) of each other as they rotate. Moreover, the fixed ends about which the rods are rotating are connected by a wire, so these ends are at the same electric potential. If a potential difference of \(4.5 \times 10^{3} \mathrm{V}\) is required to cause a \(1.0-\mathrm{mm}\) spark in air, what is the angular speed (in rad/s) of the rods when a spark jumps across the gap?

Short Answer

Expert verified
The angular speed is approximately 4141.92 rad/s.

Step by step solution

01

Understand the problem

Two conducting rods rotate in opposite directions about fixed ends which are connected by a wire, placing them at the same potential. The rods rotate perpendicularly to a magnetic field and we must find the angular speed at which the induced potential difference causes a spark across a gap between the rod ends.
02

Know the formula for potential difference in rotating rods

The potential difference between the ends of rotating rods in a magnetic field is given by:\[V = \frac{1}{2} B L^2 \omega\]where \(B\) is the magnetic field strength, \(L\) is the length of the rod, and \(\omega\) is the angular speed of the rod.
03

Set up the equation for angular speed

Set the known potential difference equal to the formula: \[4.5 \times 10^{3} = \frac{1}{2} \times 4.7 \times (0.68)^2 \times \omega\] and solve for \(\omega\).
04

Solve for angular speed \(\omega\)

Re-arrange the formula to solve for \(\omega\):\[\omega = \frac{2 \times 4.5 \times 10^{3}}{4.7 \times (0.68)^2}\]Calculate the value to find the angular speed.
05

Final calculation

Perform the calculation:\[\omega = \frac{9000}{4.7 \times 0.4624} \\omega \approx \frac{9000}{2.17248} \\omega \approx 4142.36 \, \text{rad/s}.\] Thus, the angular speed required is approximately 4141.92 rad/s.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rotating Conductors
In electromagnetic induction, rotating conductors play a vital role. Imagine two metallic rods spinning around different axes. They move through a magnetic field, chopping it as they turn. This movement is key in generating an electromotive force (EMF) across the rods.

A rotating rod in a magnetic field experiences a change in magnetic flux. This change generates a voltage along the rod, a process known as electromagnetic induction. As these rods turn in opposite directions, they create differing potential ends. This potential difference can be harnessed for various applications, such as lighting up lamps or even triggering electric sparks.
Magnetic Fields
Magnetic fields are invisible forces that exert influence on moving charges and magnetic materials. They are depicted as lines passing from the north to the south pole of a magnet.

In our example, the conducting rods were moving perpendicularly to a magnetic field of strength 4.7 Tesla. Tesla is the unit used for magnetic field strength.
  • Magnetic field lines closer are stronger.
  • Perpendicular motion to such fields maximizes this induced potential difference.

Understanding magnetic fields is critical for comprehending how sparks are generated by rotating conductors.
Potential Difference
The potential difference, commonly known as voltage, is the electrical tension between two points. This tension drives current. When it comes to our conducting rods, the potential difference is essential. It's what leads to the ultimate spark.

Using the formula for rotating rods, \[V = \frac{1}{2} B L^2 \omega\] where \(V\) is the potential difference, \(B\) is magnetic field strength, \(L\) is the rod length, and \(\omega\) is angular speed, we can compute how much the rods' potential varies.
  • High potential differences are needed to create visible sparks.
  • In our example, it was 4500 V for a 1 mm spark.

This difference in potential creates energy capable of being unleashed as sparks.
Angular Speed
Angular speed measures how quickly an object rotates, usually in radians per second (rad/s). For our rods, angular speed determines how fast they must spin to produce a voltage high enough to make a spark.

Re-arranging our earlier equation, you find the angular speed needed for our scenario:\[\omega = \frac{2 \times 4.5 \times 10^{3}}{4.7 \times (0.68)^2} \approx 4142.36\,\text{rad/s}\]
  • This tells us how fast each rod needs to rotate.
  • Achieving such speeds ensures the electrical potential demands are met.
Experiencing high angular speed results in a more potent potential difference across the conductors.
Electric Sparks
Electric sparks are fascinating physical phenomenons that occur when a sufficient voltage causes a break down in air's resistance, allowing electricity to passage freely for a brief moment. This brief free flow of electrons creates visible light, known as a spark.

In the rod scenario, the rods come within 1 mm, and the potential difference is large enough (4500 volts) to jump this gap, resulting in a spark.
  • Sparks can cause ignition in flammable atmospheres.
  • They are clear visual indicators of significant electrical power.
  • In practical applications, sparks need carefully managed to ensure safety.
Understanding sparks enhances comprehension of energy conversion in electrical systems.

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Most popular questions from this chapter

The drawing shows a type of flow meter that can be used to measure the speed of blood in situations when a blood vessel is sufficiently exposed (e.g., during surgery). Blood is conductive enough that it can be treated as a moving conductor. When it flows perpendicularly with respect to a magnetic field, as in the drawing, electrodes can be used to measure the small voltage that develops across the vessel. Suppose that the speed of the blood is 0.30 m/s and the diameter of the vessel is 5.6 mm. In a 0.60-T magnetic field what is the magnitude of the voltage that is measured with the electrodes in the drawing?

A constant current of \(I=15\) A exists in a solenoid whose inductance is \(L=3.1 \mathrm{H}\) . The current is then reduced to zero in a certain amount of time. \((\mathrm{a})\) If the current goes from 15 to 0 \(\mathrm{A}\) in a time of \(75 \mathrm{ms},\) what is the emf induced in the solenoid? (b) How much electrical energy is stored in the solenoid? (c) At what rate must the electrical energy be removed from the solenoid when the current is reduced to 0 \(\mathrm{A}\) in a time of 75 \(\mathrm{ms}\) ? Note that the rate at which energy is removed is the power.

In each of two coils the rate of change of the magnetic flux in a single loop is the same. The emf induced in coil 1 , which has 184 loops, is 2.82 \(\mathrm{V}\) . The emf induced in coil 2 is 4.23 \(\mathrm{V}\) . How many loops does coil 2 have?

A flat circular coil with 105 turns, a radius of \(4.00 \times 10^{-2} \mathrm{m}\) and a resistance of 0.480\(\Omega\) is exposed to an external magnetic field that is directed perpendicular to the plane of the coil. The magnitude of the external magnetic field is changing at a rate of \(\Delta B / \Delta t=0.783 \mathrm{T} / \mathrm{s}\) , thereby inducing a current in the coil. Find the magnitude of the magnetic field at the center of the coil that is produced by the induced current.

A copper rod is sliding on two conducting rails that make an angle of 19 with respect to each other, as in the drawing. The rod is moving to the right with a constant speed of 0.60 \(\mathrm{m} / \mathrm{s} .\) A \(0.38-\mathrm{T}\) uniform magnetic field is perpendicular to the plane of the paper. Determine the magnitude of the average emf induced in the triangle \(A B C\) during the 6.0 -s period after the rod has passed point \(A\) .
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