An infinitely long and thin wire carrying current I1 in the...

An infinitely long and thin wire carrying current $I_1$ in the negative ydirection lies along the $\mathrm{y}$-axis $(x=0, z=0)$. Another short and thin wire carrying current $I_2$ in the positive x-direction lies parallel to the x-axis. Its two ends are at $(0,0, h)$ and $(l, 0, h)$. Find the force on the short wire due to the long wire.

Key Concepts

Magnetic Field of a Long Straight Conductor

This concept explains that a long, straight wire carrying a steady current produces a magnetic field that encircles the wire. Its magnitude decreases inversely with distance from the wire, and its direction can be determined using Ampère’s law or the right-hand rule. The field is given by the formula B = (??I)/(2?r), where ?? is the permeability of free space, I is the current, and r is the radial distance from the wire.

Lorentz Force on a Current-Carrying Conductor

This concept outlines that a current-carrying conductor placed in an external magnetic field experiences a force. The force on a segment of the conductor is given by F = I (L × B), where I is the current, L is a vector representing the length and direction of the current element, and B is the magnetic field. The direction of the force is determined by the cross product and can be found using the right-hand rule.

Biot-Savart Law

The Biot-Savart Law is a fundamental principle used to calculate the magnetic field generated by a small segment of current. It states that the differential magnetic field dB at a point in space is proportional to the current element and inversely proportional to the square of the distance from the element, with the direction given by the cross product of the current element vector and the unit vector pointing from the element to the field point. This law is essential for understanding the magnetic field contributions from complex current configurations.

Right-Hand Rule

The right-hand rule is a mnemonic used to determine the direction of magnetic fields around current-carrying conductors and the direction of the force experienced by a current element in a magnetic field. When applied, it involves orienting the thumb of the right hand in the direction of the current and the fingers in the direction of the magnetic field, with the resulting force being in the direction of the palm. This rule is crucial for visualizing vector directions in electromagnetic interactions.

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