UN 2008 Physics: Magnetic Induction At Point Q Explained

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Hey guys! Let's dive into a classic physics problem from the UN 2008 exam. We're going to break down how to calculate the magnetic induction at a specific point caused by a current-carrying wire. This kind of problem often pops up, so understanding the concepts is super important. We'll be looking at the direction and magnitude of the magnetic field. Grab your physics books, and let's get started!

The Problem: Setting the Stage

Okay, so the scenario is this: We have a straight wire carrying an electric current. This wire is placed in a specific configuration, and we're given the current's magnitude. The problem specifically asks about the magnetic induction at point Q. Remember, magnetic induction is also known as the magnetic flux density, and it tells us how strong the magnetic field is at a particular point. The key here is using the right formula and applying the right-hand rule to figure out the direction. These kinds of problems are designed to test your understanding of how magnetic fields are created by moving charges (in this case, the current in the wire) and your ability to visualize the field's behavior in space. The units and constants can sometimes be a bit tricky, so always double-check. The magnetic field's strength depends on how much current is flowing through the wire, and how far away you are from the wire. These kinds of calculations are fundamental to lots of practical applications, from electric motors to magnetic resonance imaging (MRI) machines. Basically, understanding the principles helps you understand the 'why' behind a lot of technology. Let's start with the basics.

Understanding the Fundamentals: Current and Magnetic Fields

Before we jump into the calculation, let's refresh our understanding of the relationship between electric current and magnetic fields. When electric current flows through a wire, it generates a magnetic field around it. This is a fundamental concept in electromagnetism, and it's something that Michael Faraday first observed. The magnetic field lines form concentric circles around the wire. The direction of the magnetic field is determined by the right-hand rule: If you point your thumb in the direction of the current, your fingers curl in the direction of the magnetic field. The strength of the field decreases as you move further away from the wire. The key is to remember that the magnetic field is a vector quantity; that means it has both magnitude and direction. We will use the Biot-Savart Law, or simplified version, for straight wires to calculate the magnitude of the magnetic field produced by a straight current-carrying wire, which will be the basis of this calculation.

The Formula: Putting it into Practice

Alright, time to get our hands dirty with the formulas! The magnitude of the magnetic field (B) due to a long, straight wire carrying current is given by:

  • B = (μ₀ * I) / (2π * r)*

Where:

  • B is the magnetic field strength (in Tesla, T)
  • μ₀ is the permeability of free space (4π x 10⁻⁷ T⋅m/A) This is an important constant to remember!
  • I is the current flowing through the wire (in Amperes, A)
  • r is the perpendicular distance from the wire to the point where you're measuring the field (in meters, m)

Now, let's apply this to the problem. We're given that the current (I) is 7A, and we need to determine the distance (r) from the wire to point Q. We also know μ₀ from the constant. We will plug the numbers into this equation and do the math. Remember that the distance must be in meters, so make sure to convert any other units properly. The hardest part of these questions is keeping track of the units, which can trip you up in the exam, so be careful. Another thing to consider is how many wires are there. Because in the question we only have one, the calculation becomes a bit easier. It's really just a plug-and-chug problem if you know the formula and the givens.

Step-by-Step Calculation: Finding the Magnetic Field at Point Q

Let's assume, for the sake of the problem, that the distance r (from the wire to point Q) is 0.1 meters. Now let's calculate the magnetic field strength. So using the formula above, the calculation goes as follows:

  • B = (4π x 10⁻⁷ T⋅m/A * 7A) / (2π * 0.1 m)
  • B = (28π x 10⁻⁷ T⋅m) / (0.2π m)
  • B = 14 x 10⁻⁶ T

So, the magnitude of the magnetic field at point Q is approximately 14 x 10⁻⁶ Tesla. See, not so bad, right?

Determining the Direction: Right-Hand Rule in Action

Now, let's talk about the direction. This is where the right-hand rule comes into play. Imagine you're holding the wire with your right hand. Your thumb points in the direction of the current (which we are told), and your fingers curl in the direction of the magnetic field lines. Now think about point Q in relation to the wire. Point Q is located around the wire. Therefore the magnetic field will be tangential to the circle. Based on the position of point Q, and the direction of the current, we can determine the direction of the magnetic field at that point. Because magnetic fields form closed loops, the field lines will circle around the wire. If the current is going 'up' in the diagram, and point Q is to the right of the wire, the magnetic field will be directed either into the page or out of the page. This is usually determined by where the point Q is located. With some knowledge, this part of the calculation becomes easier.

Applying the Right-Hand Rule: Directional Analysis

Here's how to apply the right-hand rule:

  1. Point your thumb in the direction of the current in the wire (let's say upwards, or as the question indicates).
  2. Curl your fingers. Your fingers will curl around the wire in the direction of the magnetic field lines.
  3. At point Q, the direction of the magnetic field will be perpendicular to the line connecting the wire and point Q, and either into or out of the plane of the paper. Without a visual diagram, you need some more info to determine the final direction.

Since this is an exam question, we have a few options to choose from. Let's assume point Q is located in a way that the magnetic field at Q points directly into the plane of the paper (or the screen you are reading on). The field lines circle the wire, so at a specific point, it will have a specific direction. Always remember that the magnetic field is perpendicular to both the current and the distance from the wire.

Conclusion: Putting it All Together

Alright, let's recap. We have:

  • Calculated the magnitude of the magnetic field at point Q using the formula.
  • Determined the direction of the magnetic field using the right-hand rule.

So, your final answer would be the magnitude you calculated, combined with the direction. This direction will be either into the page or out of the page (or some other direction, depending on the setup of the problem). By breaking it down step by step, you can confidently solve problems like these, even when dealing with other arrangements of wires or more complex scenarios. These principles are key to understanding the broader concepts of electromagnetism and its applications in our daily lives.

Tips and Tricks: Ace the Physics Exam!

Here are some extra tips to help you succeed on your physics exams:

  • Practice, practice, practice! The more problems you solve, the more comfortable you'll become with the formulas and concepts.
  • Draw diagrams! Visualizing the problem with diagrams can make it much easier to understand and solve.
  • Pay attention to units! Make sure all your units are consistent before plugging them into a formula.
  • Know your constants! Memorize the important constants, like the permeability of free space (μ₀).
  • Master the right-hand rule! This is essential for determining the direction of magnetic fields.

Good luck with your exams, guys! Keep practicing, and you'll do great! And remember, physics is all about understanding the world around us. So, enjoy the journey!