Parallel conducting tracks, separated by 2.0 cm, run north and south. There is a uniform magnet

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Parallel conducting tracks, separated by 2.0 cm, run north...

Parallel conducting tracks, separated by $2.0 \mathrm{~cm}$, run north and south. There is a uniform magnetic field of 1.2 T pointing upward (out of the page). A $0.040-\mathrm{kg}$ cylindrical metal rod is placed across the tracks and a battery is connected between the tracks, with its positive terminal connected to the east track. If the current through the rod is $3.0 \mathrm{~A}$, find the magnitude and direction of the magnetic force on the rod.

Key Concepts

Magnetic Force on a Current-Carrying Conductor

This concept describes the force experienced by a conductor that carries an electric current when it is situated in a magnetic field. The force is determined by the equation F = I(L × B), where I is the current, L is the vector representing the length and direction of the conductor, and B is the magnetic field vector. This principle underlies many devices in electromagnetism and is crucial for understanding how electrical energy can be converted into mechanical work.

Right-Hand Rule

The right-hand rule is a mnemonic used to determine the direction of the resultant force when a current-carrying conductor is placed in a magnetic field. By pointing the thumb of the right hand in the direction of the conventional current, and the index finger in the direction of the magnetic field, the middle finger will point in the direction of the force acting on the conductor. This helps in visualizing the vector cross product inherent in the F = I(L × B) formula.

Uniform Magnetic Field

A uniform magnetic field is one in which both the magnitude and the direction of the field remain constant over the region of interest. This uniformity simplifies calculations in electromagnetism because the magnetic field does not vary along the length of the conductor, allowing for straightforward use of the F = I(L × B) formula without additional integration or consideration of field gradients.

Cross Product in Vector Calculus

The cross product is a fundamental operation in vector calculus that takes two vectors and produces a third vector perpendicular to both. In the context of electromagnetic forces, the cross product of the current vector (along the conductor) and the magnetic field vector determines both the magnitude and direction of the magnetic force, according to the sine of the angle between the current and magnetic field.

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