Charge Speed In Magnetic Field: Step-by-Step Guide

Hey there, physics enthusiasts! Ever wondered how fast a charged particle zips through a magnetic field? Well, buckle up, because we're about to dive into an electrifying problem that'll help us unravel this mystery. We've got a charged particle dancing in a magnetic field, and we're on a mission to figure out its speed. So, let's put on our thinking caps and get ready to solve this fascinating physics puzzle!

The Magnetic Force on a Moving Charge: An In-Depth Exploration

Let's kick things off by understanding the magnetic force acting on our charged particle. You see, when a charge ventures into a magnetic field, it experiences a force that's directly proportional to the charge's magnitude ( extit{q}), its velocity ( extit{v}), the strength of the magnetic field ( extit{B}), and the sine of the angle ( extit{θ}) between the velocity and the magnetic field. This relationship is beautifully captured by the formula: F = qvB sin θ. This equation is the cornerstone of our problem-solving journey, so let's break it down a bit further.

Think of it like this: the magnetic force is like a friendly nudge that a charged particle feels when it enters a magnetic field party. The stronger the charge ( extit{q}), the bigger the invite, and the more forceful the nudge. Similarly, the faster the particle moves ( extit{v}), the more enthusiastic the greeting, resulting in a stronger force. The intensity of the magnetic field ( extit{B}) also plays a crucial role; a more powerful field means a more vigorous interaction. And finally, the angle ( extit{θ}) adds a touch of directionality. If the particle moves parallel to the magnetic field (θ = 0° or 180°), it's like a wallflower at the party, experiencing no force at all (sin 0° = sin 180° = 0). But if it moves perpendicularly (θ = 90°), it's right in the thick of the action, feeling the full force (sin 90° = 1).

Now, let's connect this theory to our specific problem. We have a charge of 8.4 × 10⁻⁴ Coulombs (C) cruising through a magnetic field of 6.7 × 10⁻³ Tesla (T). The angle between the charge's velocity and the magnetic field is 35°, and the resulting magnetic force is a hefty 3.5 × 10⁻² Newtons (N). Our mission, should we choose to accept it, is to determine the velocity ( extit{v}) of this charged particle. We've got all the pieces of the puzzle; now it's time to put them together.

To find the velocity, we need to rearrange our trusty formula. We'll do a bit of algebraic magic, isolating extit{v} on one side of the equation. This gives us: v = F / (qB sin θ). This equation is our key to unlocking the velocity. We've got the force (F), the charge (q), the magnetic field (B), and the angle (θ). All that's left is to plug in the values and let the math do its thing.

So, let's take a moment to appreciate the elegance of this equation. It beautifully encapsulates the interplay between charge, velocity, magnetic field, and force. It's a testament to the power of physics to describe the world around us. And it's the perfect tool for solving our problem. We're about to witness how a simple formula can reveal the speed of a charged particle dancing in a magnetic field.

The Calculation: Cracking the Velocity Code

Alright, guys, it's time to roll up our sleeves and get our hands dirty with some calculations. We've got our rearranged formula ready: v = F / (qB sin θ). Now, it's just a matter of plugging in the values we have: F = 3.5 × 10⁻² N, q = 8.4 × 10⁻⁴ C, B = 6.7 × 10⁻³ T, and θ = 35°. Let's substitute these into the equation:

v = (3.5 × 10⁻² N) / ((8.4 × 10⁻⁴ C) × (6.7 × 10⁻³ T) × sin(35°))

Now, let's break this down step by step to make sure we don't miss anything. First, we need to calculate the sine of 35°. Using a calculator, we find that sin(35°) ≈ 0.5736. Next, we'll multiply the values in the denominator:

(8.4 × 10⁻⁴ C) × (6.7 × 10⁻³ T) × 0.5736 ≈ 3.23 × 10⁻⁶ C·T

Remember, the units are crucial! We're dealing with Coulombs (C) for charge and Tesla (T) for the magnetic field strength. Now, we can substitute this result back into our equation:

v = (3.5 × 10⁻² N) / (3.23 × 10⁻⁶ C·T)

Finally, we perform the division to find the velocity:

v ≈ 1.08 × 10⁴ m/s

Wow! That's a pretty impressive speed. Our charged particle is zooming through the magnetic field at approximately 1.08 × 10⁴ meters per second. That's about 10,800 meters every second, which is seriously fast! To put it in perspective, that's like covering over 10 kilometers in a single second.

So, what does this number tell us? Well, it gives us a concrete understanding of how quickly the charge is moving under the influence of the magnetic force. It highlights the powerful effect that magnetic fields can have on moving charges. And it demonstrates the precision with which we can calculate these interactions using physics principles.

But beyond the numerical answer, this calculation is a reminder of the beauty and power of physics. We took a seemingly complex scenario – a charged particle moving through a magnetic field – and broke it down into manageable pieces. We applied a fundamental equation, plugged in the values, and arrived at a meaningful result. It's a testament to the elegance and predictive power of physics. And it's just plain cool!

Interpreting the Results: Velocity in Context

Now that we've crunched the numbers and found the velocity of our charged particle (approximately 1.08 × 10⁴ m/s), let's take a moment to put this result into context. What does this speed actually mean? How does it compare to other speeds we encounter in the world around us? And what are the implications of this high velocity?

First, let's acknowledge just how fast 1.08 × 10⁴ m/s really is. As we mentioned earlier, this is about 10,800 meters per second. To put that in perspective, the speed of sound in air is roughly 343 meters per second. So, our charged particle is moving over 30 times faster than the speed of sound! That's supersonic and then some.

Another way to visualize this speed is to compare it to the speed of a typical airplane. Commercial airliners usually cruise at speeds around 250 meters per second. Our charged particle is zipping along at more than 40 times the speed of a jet plane. Imagine trying to keep up with that!

But why is this particle moving so fast? The answer lies in the interplay between the magnetic force and the particle's charge and mass. The magnetic force is acting as a constant push on the charge, accelerating it to high speeds. The stronger the magnetic field and the greater the charge, the more forceful the push and the faster the particle accelerates. Conversely, the more massive the particle, the more resistant it is to acceleration, and the lower its final velocity will be.

This concept of charged particles moving at high speeds in magnetic fields is not just a theoretical curiosity; it has practical applications in a wide range of technologies. For example, particle accelerators, like the Large Hadron Collider (LHC) at CERN, use powerful magnetic fields to accelerate charged particles to incredibly high speeds, close to the speed of light. These high-speed particles are then collided, allowing physicists to study the fundamental building blocks of matter.

Magnetic Resonance Imaging (MRI), a crucial tool in modern medicine, also relies on the interaction of magnetic fields and charged particles. MRI scanners use strong magnetic fields to align the nuclear spins of hydrogen atoms in the body. Radio waves are then used to disrupt this alignment, and the signals emitted by the atoms as they return to their equilibrium state are used to create detailed images of the body's internal structures. The precise control and manipulation of charged particles in magnetic fields are essential for the functioning of MRI technology.

Furthermore, the Earth itself has a magnetic field that plays a vital role in protecting us from harmful solar radiation. Charged particles emitted by the Sun, known as the solar wind, are deflected by the Earth's magnetic field, preventing them from reaching the surface and potentially causing damage to our atmosphere and electronic systems. This natural magnetic shield is a crucial factor in making our planet habitable.

In conclusion, the high velocity of our charged particle in the magnetic field is not just an isolated result; it's a window into a world of fascinating physics and practical applications. It highlights the power of magnetic fields to influence the motion of charged particles, and it underscores the importance of these interactions in technologies that shape our modern world.

Conclusion: The Elegance of Electromagnetism

So, there you have it, guys! We've successfully navigated the world of electromagnetism and calculated the velocity of a charged particle zipping through a magnetic field. We took a real-world physics problem, dissected it, applied the relevant formula, and arrived at a meaningful answer. It's a testament to the power of physics to explain and predict the behavior of the universe around us.

We started by understanding the fundamental principle: the magnetic force on a moving charge. We learned how this force depends on the charge's magnitude, its velocity, the magnetic field strength, and the angle between the velocity and the field. We then translated this understanding into a mathematical equation: F = qvB sin θ. This equation became our compass, guiding us through the problem-solving process.

Next, we rearranged the equation to isolate the variable we were interested in: the velocity, v. This algebraic maneuver gave us the tool we needed to crack the problem: v = F / (qB sin θ). With our formula in hand, we carefully plugged in the given values, mindful of units and significant figures. We performed the calculations step-by-step, ensuring accuracy and clarity.

The result? A velocity of approximately 1.08 × 10⁴ m/s. This number, on its own, is impressive. But we didn't stop there. We took the time to interpret this result, to put it into context. We compared it to familiar speeds, like the speed of sound and the speed of an airplane. We explored the implications of this high velocity and its relevance to various technologies, from particle accelerators to MRI scanners.

Throughout this journey, we've not only solved a physics problem but also gained a deeper appreciation for the elegance and power of electromagnetism. We've seen how a few fundamental principles can explain a wide range of phenomena, from the motion of charged particles to the functioning of medical imaging devices. We've witnessed the interplay between theory and experiment, between abstract equations and concrete applications.

And perhaps most importantly, we've reinforced the idea that physics is not just a collection of formulas and equations; it's a way of thinking about the world. It's a process of asking questions, making observations, building models, and testing predictions. It's a journey of discovery, driven by curiosity and guided by logic.

So, the next time you see a magnet attracting a piece of metal, or hear about the latest advances in medical imaging, remember the principles we've discussed here. Remember the magnetic force, the velocity of charged particles, and the power of electromagnetism. And remember that you, too, can unlock the secrets of the universe, one physics problem at a time.

Original Question: If a charge of $8.4 imes 10^{-4} C$ moves at an angle of $35^{\circ}$ to a magnetic field that has a field strength of $6.7 imes 10^{-3} T$ and the magnetic force is $3.5 imes 10^{-2} N$, what is the speed of the charge?

Revised Question: A charged particle with a magnitude of $8.4 imes 10^{-4}$ Coulombs moves at an angle of $35$ degrees relative to a magnetic field. The magnetic field has a strength of $6.7 imes 10^{-3}$ Tesla, and the resulting magnetic force on the particle is $3.5 imes 10^{-2}$ Newtons. Determine the speed of the moving charge.

Charge Speed in a Magnetic Field: Step-by-Step Solution