(II) A current-carrying circular loop of wire (radius r, current I) is partially immersed in a magnetic field of constant magnitude B₀ directed out of the page as shown in Fig. 27–43. Determine the net force on the loop due to the field in terms of θ₀. (Note that θ₀ points to the dashed line, above which B = 0.)

Verified step by step guidance

1. Understand the problem: A circular loop of wire carrying a current I is partially immersed in a uniform magnetic field B₀. The goal is to determine the net force on the loop due to the magnetic field in terms of the angle θ₀, which defines the portion of the loop in the field.
2. Recall the formula for the magnetic force on a current-carrying wire: The force on a small segment of the wire is given by **dF = I (dL × B)**, where dL is the infinitesimal length vector of the wire and B is the magnetic field vector. The cross product ensures that the force is perpendicular to both dL and B.
3. Break the loop into segments: The loop is symmetric, so the forces on opposite segments of the loop will cancel out in certain directions. Focus on the vertical components of the force, as the horizontal components will cancel due to symmetry.
4. Integrate the force over the portion of the loop in the magnetic field: The magnetic field is present only in the region defined by the angle θ₀. The force on a small segment of the loop can be expressed as **dF = I B₀ r dθ**, where r is the radius of the loop and dθ is the angular width of the segment. The vertical component of the force is **dF_y = dF sin(θ)**.
5. Set up and evaluate the integral: To find the net vertical force, integrate the vertical component of the force over the range of angles where the magnetic field is present (from -θ₀ to θ₀). The integral is **F_net = ∫_{-θ₀}^{θ₀} I B₀ r sin(θ) dθ**. Solve this integral to express the net force in terms of θ₀, I, B₀, and r.

Key Concepts

Magnetic Force on a Current-Carrying Wire

When a current-carrying wire is placed in a magnetic field, it experiences a magnetic force. This force is given by the equation F = I(L × B), where F is the force, I is the current, L is the length vector of the wire segment in the field, and B is the magnetic field vector. The direction of the force can be determined using the right-hand rule, which states that if you point your thumb in the direction of the current and your fingers in the direction of the magnetic field, your palm will face the direction of the force.

Lorentz Force Law

The Lorentz force law describes the force experienced by a charged particle moving in an electromagnetic field. It combines both electric and magnetic forces and is expressed as F = q(E + v × B), where F is the total force, q is the charge, E is the electric field, v is the velocity of the charge, and B is the magnetic field. In the context of a current-carrying loop, the law helps to analyze how different segments of the loop interact with the magnetic field, leading to a net force.

Magnetic Field and Its Direction

A magnetic field is a vector field that exerts a force on moving charges and magnetic dipoles. The direction of the magnetic field is defined by the orientation of the field lines, which indicate the direction a north pole of a magnet would move. In this problem, the magnetic field B₀ is directed out of the page, and understanding its direction is crucial for determining how the current in the loop interacts with the field, particularly in calculating the net force on the loop.