(III) A curved wire, connecting two points a and b, lies in a plane perpendicular to a uniform magnetic field $\overrightarrow{B}$ and carries a current I. Show that the resultant magnetic force on the wire, no matter what its shape, is the same as that on a straight wire connecting the two points carrying the same current I. See Fig. 27–44.

Verified step by step guidance

1. The magnetic force on a current-carrying wire in a magnetic field is given by the formula: **F = I ∫ (dl × B)**, where **dl** is the infinitesimal vector element of the wire, **B** is the magnetic field vector, and **×** represents the cross product.
2. The key observation here is that the magnetic field **B** is uniform and constant in magnitude and direction. This allows us to treat **B** as a constant vector throughout the integration process.
3. The integral **∫ (dl × B)** can be rewritten as **B × ∫ dl**, because the cross product with a constant vector can be factored out of the integral. This simplifies the problem to evaluating the integral of **dl**, which represents the vector sum of all infinitesimal elements of the wire.
4. The vector sum **∫ dl** is simply the displacement vector from point **a** to point **b**, regardless of the shape of the wire. This is because the integral of **dl** over a path gives the net displacement between the start and end points.
5. Thus, the magnetic force becomes **F = I (B × d)**, where **d** is the straight-line displacement vector from **a** to **b**. This shows that the resultant magnetic force depends only on the endpoints of the wire and not on its shape, proving that the force on the curved wire is the same as that on a straight wire connecting the two points.

Key Concepts

Magnetic Force on a Current-Carrying Wire

The magnetic force experienced by a current-carrying wire in a magnetic field is given by the equation F = I(L × B), where F is the force, I is the current, L is the length vector of the wire, and B is the magnetic field vector. This force is perpendicular to both the direction of the current and the magnetic field, resulting in a force that can vary depending on the wire's orientation and shape.

Biot-Savart Law

The Biot-Savart Law describes the magnetic field generated by a current-carrying conductor. It states that the magnetic field dB at a point in space due to a small segment of current-carrying wire is proportional to the current and the length of the segment, and inversely proportional to the square of the distance from the segment to the point. This law helps in understanding how the shape of the wire affects the resultant magnetic field and force.

Superposition Principle

The superposition principle in physics states that the total force acting on an object is the vector sum of all individual forces acting on it. In the context of a curved wire in a magnetic field, this principle allows us to analyze the magnetic forces on infinitesimal segments of the wire and sum them up to find the total force, demonstrating that the resultant force on a curved wire can be equivalent to that on a straight wire connecting the same endpoints.