Magnetic Force On A Conductor: Step-by-Step Calculation

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Magnetic Force on a Conductor: Step-by-Step Calculation

Let's dive into calculating the magnetic force on a copper conductor! This is a classic physics problem that combines electromagnetism principles. We'll break it down step-by-step to make sure you understand every part of the process. So, grab your calculators and let's get started!

Understanding the Problem

First, let's recap the scenario. We have a 60cm copper conductor sitting pretty on the x-axis. This conductor is hooked up to a 20V electromotive force (EMF), which is basically a voltage source. Now, here's where it gets interesting: the conductor is chilling in a uniform magnetic field with a strength of 1.5T (that's Tesla, the unit for magnetic field strength). This magnetic field is pointing along the positive z-axis. Our mission, should we choose to accept it, is to figure out the magnetic force vector acting on the conductor.

Breaking Down the Components

Before we jump into calculations, let's clarify a few key concepts:

  • Conductor Length (L): The length of our copper wire, which is 60cm or 0.6 meters.
  • Electromotive Force (EMF or V): The voltage supplied to the conductor, which is 20V.
  • Magnetic Field Strength (B): The strength of the magnetic field, which is 1.5T.
  • Direction of Magnetic Field: This is crucial! The magnetic field is pointing along the positive z-axis.

Why This Matters

You might be wondering, "Why should I care about the magnetic force on a conductor?" Well, this principle is the backbone of many technologies we use every day! Think about electric motors, loudspeakers, and magnetic levitation trains. They all rely on the interaction between electric currents and magnetic fields to generate forces. Understanding this concept is fundamental to grasping how these devices work.

Calculating the Magnetic Force

Now, let's get our hands dirty with some calculations. The magnetic force on a current-carrying conductor in a magnetic field is given by the following formula:

F = I * L * B

Where:

  • F is the magnetic force vector.
  • I is the current flowing through the conductor.
  • L is the length vector of the conductor (magnitude is the length, direction is the current direction).
  • B is the magnetic field vector.

Finding the Current (I)

We know the voltage (V) and we need to find the current (I). To do this, we'll use Ohm's Law:

V = I * R

Where:

  • V is the voltage (20V).
  • I is the current (what we want to find).
  • R is the resistance of the copper conductor.

But wait! We don't know the resistance (R) of the copper conductor. To find it, we need more information, such as the resistivity of copper and the cross-sectional area of the wire. Let's assume we somehow know (or calculate using additional data not provided) that the resistance of the wire is, say, 0.1 ohms. This is a crucial piece of information that's missing from the original problem statement. Without it, we can't proceed to a numerical answer. Always remember to check if you have all the necessary information before trying to solve a physics problem! Let's proceed assuming we have this resistance value.

Now we can find the current:

I = V / R = 20V / 0.1 ohms = 200A

So, the current flowing through the conductor is 200 Amperes.

Determining the Length Vector (L)

The length vector L has a magnitude equal to the length of the conductor (0.6 meters) and a direction along the x-axis (since the conductor is placed along the x-axis). Therefore, we can write L as:

L = 0.6i

Where i is the unit vector along the x-axis.

Calculating the Magnetic Force Vector (F)

Now we have all the pieces we need! Let's plug the values into the magnetic force formula:

F = I * L x B

Remember that the "x" here represents the cross product, not simple multiplication. The cross product of two vectors results in another vector that is perpendicular to both original vectors. This is where the direction of the magnetic force comes from.

We have:

  • I = 200A
  • L = 0.6i
  • B = 1.5k

Where k is the unit vector along the z-axis.

So,

F = 200 * (0.6i) x (1.5k)

F = 200 * 0.6 * 1.5 * (i x k)

F = 180 * (i x k)

Now, we need to remember the cross product rules for unit vectors. Specifically, i x k = -j. Therefore:

F = 180 * (-j)

F = -180j

Interpreting the Result

The magnetic force vector is -180j Newtons. This means:

  • The magnitude of the force is 180 Newtons.
  • The direction of the force is along the negative y-axis.

So, the copper conductor experiences a force of 180 Newtons pushing it in the negative y-direction.

Putting it All Together

Let's recap what we did:

  1. We understood the problem and identified the given information.
  2. We found the current flowing through the conductor using Ohm's Law (after assuming a resistance value).
  3. We determined the length vector of the conductor.
  4. We calculated the magnetic force vector using the formula F = I * L x B and the cross product.
  5. We interpreted the result to understand the magnitude and direction of the force.

Importance of Units

Always, always, always pay attention to units! Make sure all your values are in consistent units (meters for length, Amperes for current, Tesla for magnetic field, etc.) before plugging them into formulas. Using the wrong units will lead to incorrect results.

Common Mistakes to Avoid

  • Forgetting the Cross Product: The magnetic force involves the cross product, not simple multiplication. Make sure you understand how to calculate the cross product correctly.
  • Ignoring the Direction: The direction of the magnetic field and current are crucial. Use the right-hand rule to determine the direction of the force if needed.
  • Using Incorrect Units: Double-check your units to make sure they are consistent.
  • Missing Information: As highlighted in this example, sometimes you won't be given all the information you need directly. You might need to use other formulas or look up material properties (like the resistivity of copper) to find the missing pieces.

Expanding Your Knowledge

Want to learn more about magnetic forces? Here are some topics to explore:

  • Right-Hand Rule: A handy tool for determining the direction of the magnetic force.
  • Magnetic Fields due to Currents: How electric currents create magnetic fields.
  • Lorentz Force: The total force on a charged particle in electric and magnetic fields.
  • Applications of Magnetic Forces: Electric motors, generators, magnetic levitation, and more!

Conclusion

Calculating the magnetic force on a conductor might seem daunting at first, but by breaking it down into smaller steps and understanding the underlying principles, it becomes much more manageable. Remember to pay attention to units, use the correct formulas, and practice, practice, practice! Now go forth and conquer the world of electromagnetism!

Disclaimer: This explanation assumes a resistance value for the copper conductor, which was not provided in the original problem statement. In a real-world scenario, you would need to determine the resistance using additional information.