In Fig. 28-44, a metal wire of mass m=24.1 mg can slide with negligible friction on two horizont

step 1 So here we have two wires, the green wires, and there is a metal wire of mass mass 24 .1 milligr

In Fig. 28-44, a metal wire of mass m=24.1 mg can slide with...

In Fig. $28-44$, a metal wire of mass $m=24.1 \mathrm{mg}$ can slide with negligible friction on two horizontal parallel rails separated by distance $d=2.56 \mathrm{~cm} .$ The track lies in a vertical uniform magnetic field of magnitude $56.3 \mathrm{mT}$. At time $t=0$, device $G$ is connected to the rails, producing a constant current $i=9.13 \mathrm{~mA}$ in the wire and rails (even as the wire moves). At $t=61.1 \mathrm{~ms}$, what are the wire's (a) speed and (b) direction of motion (left or right)?

Key Concepts

Magnetic Force on a Current-Carrying Conductor

When a current flows through a conductor placed in a magnetic field, the magnetic force acting on the conductor is given by the cross product of the current element and the magnetic field. This concept, often expressed as F = i (L × B), underpins how the conductor experiences a force whose magnitude depends on the current, the length of the conductor in the field, and the strength of the magnetic field.

Right-Hand Rule for Determining Force Direction

The direction of the force on a current-carrying conductor in a magnetic field is determined using the right-hand rule. By pointing the thumb in the direction of the conventional current and the fingers in the direction of the magnetic field, the palm points in the direction of the force. This rule is crucial for predicting the trajectory of the conductor’s motion.

Newton’s Second Law of Motion

Once the force on the conductor is established, Newton’s second law connects this force to the resulting acceleration. The law, expressed as F = m*a, allows for calculation of the acceleration and subsequent speed of the conductor over time when the force and mass are known.

Electromagnetic Induction in Moving Conductors

Even though the current is maintained constant in this scenario, moving conductors in a magnetic field can induce an electromotive force (emf) due to the change in magnetic flux. This concept is fundamental in electromagnetic systems, affecting energy balance and ensuring that induced emfs oppose the motion (as per Lenz’s law), though external power supplies can adjust to maintain constant current.

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