A long wire carrying 4.50 A of current makes two 90° bends, as shown in Fig. E27.35. The bent part of the wire passes through a uniform 0.240 T magnetic field directed as shown in the figure and confined to a limited region of space. Find the magnitude and direction of the force that the magnetic field exerts on the wire.

Verified step by step guidance

1. Identify the sections of the wire that are within the magnetic field. In the image, the wire makes two 90° bends and passes through a uniform magnetic field region. The sections of the wire within the magnetic field are the horizontal and vertical segments.
2. Use the formula for the magnetic force on a current-carrying wire: $F=ILBsin\theta$, where $I$ is the current, $L$ is the length of the wire in the magnetic field, $B$ is the magnetic field strength, and $\theta$ is the angle between the wire and the magnetic field.
3. Determine the angle $\theta$ for each segment of the wire. For the horizontal segment, the angle between the wire and the magnetic field is 90°, and for the vertical segment, the angle is 0°.
4. Calculate the force on each segment separately. For the horizontal segment, use $F=ILB$ since $sin\theta =\; 1$. For the vertical segment, the force is zero because $sin\theta =\; 0$.
5. Add the forces vectorially to find the net force on the wire. Since the force on the vertical segment is zero, the net force is equal to the force on the horizontal segment. The direction of the force can be determined using the right-hand rule, which indicates that the force is perpendicular to both the current direction and the magnetic field direction.

Key Concepts

Magnetic Force on a Current-Carrying Wire

The magnetic force on a current-carrying wire is given by the equation F = I(L × B), where I is the current, L is the length vector of the wire segment, and B is the magnetic field. The direction of the force is determined by the right-hand rule, which states that if the thumb points in the direction of the current and the fingers point in the direction of the magnetic field, the palm faces the direction of the force.

Right-Hand Rule

The right-hand rule is a mnemonic for understanding the direction of the magnetic force on a current-carrying wire. When the thumb of the right hand points in the direction of the current and the fingers point in the direction of the magnetic field, the force exerted by the magnetic field is in the direction that the palm faces. This rule helps determine the orientation of the force vector in three-dimensional space.

Magnetic Field Interaction

A magnetic field exerts a force on a wire carrying an electric current when the wire is within the field. The force depends on the current's magnitude, the wire's length within the field, and the angle between the wire and the magnetic field lines. In this scenario, the wire's bends and the uniform magnetic field influence the calculation of the net force, requiring consideration of each wire segment's orientation relative to the field.