A straight wire is aligned east-west in a region where the...

A straight wire is aligned east-west in a region where the Earth's magnetic field has magnitude $0.48 \mathrm{mT}$ and direction $72^{\circ}$ below the horizontal, with the horizontal component directed due north. The wire carries a current $I$ toward the west. The magnetic force on the wire per unit length of wire has magnitude $0.020 \mathrm{~N} / \mathrm{m}$. (a) What is the direction of the magnetic force on the wire? (b) What is the current $I$ ?

Key Concepts

Magnetic Force on a Current-Carrying Conductor

This concept explains that a conductor carrying an electric current in the presence of a magnetic field experiences a force. The force is determined by the vector cross product of the current direction and the magnetic field, and its magnitude is given by F = I L × B, where I is the current, L is the length vector of the conductor, and B is the magnetic field vector. The force is always perpendicular to both the current direction and the magnetic field.

Right-Hand Rule

The right-hand rule is a fundamental method used to determine the direction of the force on a current-carrying conductor in a magnetic field. By aligning the thumb with the direction of the current and the fingers with the magnetic field, the direction in which the palm pushes corresponds to the direction of the force. This mnemonic is crucial in predicting the orientation of the force without resorting to complex calculations.

Vector Decomposition

Vector decomposition involves breaking a vector, such as a magnetic field, into its horizontal and vertical components. This process is important in understanding and analyzing systems where the magnetic field is not aligned along a single coordinate axis. By decomposing the field, it becomes easier to apply the cross product with the current direction to determine the resulting magnetic force.

Cross Product

The cross product of two vectors is a mathematical operation that results in a third vector, which is perpendicular to the plane defined by the original vectors. In the context of magnetic forces, the cross product I L × B is used to calculate the force on a current-carrying conductor, both in magnitude and direction. The use of the cross product also emphasizes the importance of having the vectors' directions correctly defined, as the order of the vectors affects the direction of the resulting force.

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