Part of a single rectangular loop of wire with dimensions shown in the (Figure 1) is situated inside a region of uniform magnetic field of 0.770 T . The total resistance of the loop is 0.450 Ω .
Figure1 of 1

Part A
Calculate the force required to pull the loop from the field (to the right) at a constant velocity of 6.20 m/s . Neglect gravity.
Express your answer to three significant figures and include the appropriate units.
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F =

To calculate the force required to pull the loop from the magnetic field at a constant velocity, we need to use concepts from electromagnetism, particularly Faraday's Law of Induction and Lorentz forces.

### Given:
- **Magnetic field strength (B):** 0.770 T
- **Velocity (v):** 6.20 m/s
- **Resistance of the loop (R):** 0.450 Ω

Let’s assume that the width of the loop (which is the dimension perpendicular to the velocity and magnetic field) is \( L \), and the length is the part inside the magnetic field (denoted as \( l \)).

#### Step 1: Induced EMF (ε)
The electromotive force (EMF) induced in the loop as it moves through the magnetic field is given by Faraday’s Law:

\[
\varepsilon = B \cdot l \cdot v
\]

Here:
- \( B \) is the magnetic field (0.770 T),
- \( l \) is the length of the part of the loop inside the magnetic field,
- \( v \) is the velocity of the loop (6.20 m/s).

#### Step 2: Induced Current (I)
The induced current in the loop is calculated using Ohm’s Law:

\[
I = \frac{\varepsilon}{R}
\]

Substituting for \( \varepsilon \):

\[
I = \frac{B \cdot l \cdot v}{R}
\]

#### Step 3: Magnetic Force (F)
The force required to pull the loop is equal to the magnetic force acting on the current in the loop, which is given by:

\[
F = B \cdot I \cdot l
\]

Substitute the expression for \( I \):

\[
F = B \cdot \left(\frac{B \cdot l \cdot v}{R}\right) \cdot l
\]

Simplify this:

\[
F = \frac{B^2 \cdot l^2 \cdot v}{R}
\]

Now, we need to plug in the values of \( B \), \( v \), and \( R \), but to complete this, we also need the value of \( l \) (the length of the loop inside the magnetic field). Could you provide that, or should we assume it's available?

### Let me know if you need any clarifications so far!

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Here are 5 related questions that expand on the above concepts:

1. How does the force change if the velocity of the loop is doubled?
2. What happens to the induced EMF if the magnetic field strength is reduced by half?
3. How would the induced current change if the resistance of the loop is increased?
4. What is the relationship between force and the length of the wire in the magnetic field?
5. How would the force be affected if the loop was pulled at an angle instead of directly to the right?

### Tip:
In problems involving magnetic fields and motion, always keep track of how EMF, current, and force depend on velocity, magnetic field strength, and geometry!

Part of a single rectangular loop of wire with dimensions shown in the (Figure 1) is situated inside a region of uniform magnetic field of 0.770 T. The total resistance of the loop is 0.450 Ω. Calculate the force required to pull the loop from the field (to the right) at a constant velocity of 6.20 m/s. Neglect gravity.

This problem demonstrates how to calculate the force required to pull a rectangular loop of wire from a uniform magnetic field at a constant velocity of 6.20 m/s, using key principles from electromagnetism such as Faraday's Law and Lorentz Force. Given: B = 0.770 T, R = 0.450 Ω. Suitable for advanced high school or college-level physics students.

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