Angle Of Conductor Movement In Magnetic Field: Calculation

Hey guys! Today, let's dive into a fascinating physics problem that involves figuring out the ideal angle at which a copper conductor should move within a magnetic field to generate a specific electromotive force (EMF). We'll break down each component of the problem, apply the relevant formulas, and solve for the unknown angle. So, grab your thinking caps, and let's get started!

Understanding the Problem

So, the big question is: At what angle to the magnetic field lines with an induction of 0.5 T should a copper conductor with a cross-section of 0.85 mm² and a resistance of 0.04 Ohms move at a speed of 0.5 m/s so that an induced EMF of 0.35 V is generated at its ends? (resistivity)

Before we start crunching numbers, let's clarify what we're trying to achieve. We need to find the exact angle at which a copper wire should move through a magnetic field so that it produces an EMF (voltage) of 0.35 V. We're given a bunch of details: the strength of the magnetic field, how fast the wire is moving, the wire's physical characteristics (like its cross-sectional area and resistance), and, of course, the desired EMF.

Key Components

• Magnetic Induction (B): 0.5 T (Tesla). This tells us how strong the magnetic field is.
• Cross-sectional Area (A): 0.85 mm². We'll need to convert this to square meters (m²) for our calculations.
• Resistance (R): 0.04 Ohms. This is the electrical resistance of the copper conductor.
• Velocity (v): 0.5 m/s. How fast the conductor is moving through the magnetic field.
• Induced EMF (ε): 0.35 V. This is the voltage we want to generate.

The Goal

Our mission, should we choose to accept it, is to find the angle (θ) between the conductor's motion and the magnetic field lines.

Laying the Groundwork: Relevant Formulas and Concepts

To solve this problem, we need to employ a few key physics concepts and formulas. These will help us link all the given information and ultimately isolate the angle we're looking for.

Faraday's Law of Electromagnetic Induction

The cornerstone of this problem is Faraday's Law. In the context of a moving conductor, it tells us that the induced EMF (ε) is proportional to the rate of change of magnetic flux through the circuit formed by the conductor. A simplified version of this law, suitable for our scenario, is:

ε = B * l * v * sin(θ)

Where:

• ε is the induced EMF.
• B is the magnetic induction.
• l is the length of the conductor within the magnetic field.
• v is the velocity of the conductor.
• θ is the angle between the velocity vector and the magnetic field vector.

Resistance, Resistivity, Length, and Area

We also know that the resistance (R) of a conductor is related to its resistivity (ρ), length (l), and cross-sectional area (A) by the following formula:

R = ρ * (l / A)

From this, we can express the length (l) of the conductor as:

l = (R * A) / ρ

Putting It All Together

Our strategy is to first find the length ( l ) of the conductor using the resistance formula, and then substitute that value into Faraday's Law to solve for the angle (θ).

Step-by-Step Solution

Let's get our hands dirty with some calculations! We'll take it one step at a time to make sure we don't miss anything.

Step 1: Convert Units

First, we need to ensure all our units are consistent. Let's convert the cross-sectional area from mm² to m²:

A = 0.85 mm² = 0.85 × 10⁻⁶ m²

Step 2: Find the Resistivity of Copper

The resistivity (ρ) of copper is a known constant. It's approximately:

ρ = 1.7 × 10⁻⁸ Ω⋅m

Step 3: Calculate the Length of the Conductor

Now we can use the resistance formula to find the length (l) of the conductor:

l = (R * A) / ρ = (0.04 Ω * 0.85 × 10⁻⁶ m²) / (1.7 × 10⁻⁸ Ω⋅m) = 2 meters

Step 4: Apply Faraday's Law and Solve for the Angle

Now we can plug the values we have into Faraday's Law:

ε = B * l * v * sin(θ)

0. 35 V = 0.5 T * 2 m * 0.5 m/s * sin(θ)

Simplify the equation:

0. 35 = 0.5 * sin(θ)

Now, isolate sin(θ):

sin(θ) = 0.35 / 0.5 = 0.7

Finally, find the angle θ by taking the inverse sine (arcsin) of 0.7:

θ = arcsin(0.7) ≈ 44.43°

The Answer

Therefore, the copper conductor should move at an angle of approximately 44.43 degrees relative to the magnetic field lines to generate an induced EMF of 0.35 V.

Wrapping Up

So, there you have it! We successfully navigated through the problem by understanding the underlying physics, applying the correct formulas, and performing the necessary calculations. Remember, guys, physics problems often seem intimidating at first, but breaking them down into smaller, manageable steps makes them much easier to solve.

Key Takeaways:

• Faraday's Law is essential for understanding electromagnetic induction.
• Unit conversion is crucial for accurate calculations.
• Step-by-step problem-solving makes complex problems easier to handle.

Keep practicing, and you'll become a physics whiz in no time! If you have any questions or want to explore other physics problems, feel free to ask. Happy calculating!