In Fig. 29-66, a long straight wire carries a current i1=30.0 A and a rectangular loop carries

step 1 The force on a loop due to current i1 will be the forces on the two perpendicular sides of the B

In Fig. 29-66, a long straight wire carries a current...

In Fig. 29-66, a long straight wire carries a current $i_{1}=30.0 \mathrm{~A}$ and a rectangular loop carries current $i_{2}=20.0 \mathrm{~A}$. Take the dimensions to be $a=1.00 \mathrm{~cm}$, $b=8.00 \mathrm{~cm},$ and $L=30.0 \mathrm{~cm} .$ In unit-vector notation, what is the net force on the loop due to $i_{1} ?$

Key Concepts

Magnetic Field Due to a Current-Carrying Wire

This concept refers to the magnetic field produced by a steady current flowing through a long, straight conductor. According to Ampère's law (or the Biot–Savart law), the magnitude of the magnetic field created at a distance from the wire is determined by the current and decreases with increasing distance. The direction of the field is given by the right?hand rule, wrapping around the wire in concentric circles.

Magnetic Force on a Current-Carrying Conductor

A conductor carrying an electric current placed in an external magnetic field experiences a force, which is derived from the interaction between the moving charges in the current and the magnetic field. This force, described by the Lorentz force law as F = I (L × B), depends on the magnitude of the current, the length of the conductor in the field, and the direction given by the vector cross product of the current element and the magnetic field.

Vector Cross Product in Force Calculations

The vector cross product is a mathematical tool used to determine the direction of the magnetic force on a current element. In the context of electromagnetic forces, it involves taking the cross product of the current direction vector and the magnetic field vector. This operation results in a force vector that is perpendicular to both the direction of current flow and the magnetic field, which is crucial for correctly assessing the net force on the loop.

Superposition and Integration of Forces on Composite Conductors

When dealing with complex current-carrying structures like a rectangular loop, it is necessary to consider the contributions from different segments. The net force on the loop is found by evaluating and summing the magnetic forces on each segment, taking into account the directions and magnitudes of these forces. Often, symmetry or cancellation between segments simplifies the overall analysis, leading to a final net force on the composite conductor.

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