A wire, carrying a current `i`, is kept in the `x-y` plane along the curve `y = A sin ((2pi)/(lambda) x)`. A magnetic field `B` exists in the z-direction. Find the magnitude of the magnetic force on the portion of the wire between `x = 0 and x = lambda`.

Explanation

To find the magnitude of the magnetic force on the portion of the wire between \( x = 0 \) and \( x = \lambda \), we can follow these steps:

### Step 1: Understand the Given Information
We have a wire carrying a current \( i \) along the curve defined by \( y = A \sin\left(\frac{2\pi}{\lambda} x\right) \) in the \( x-y \) plane. A magnetic field \( \mathbf{B} \) exists in the \( z \)-direction.

### Step 2: Identify the Force on a Small Element of the Wire
The magnetic force \( d\mathbf{F} \) on a small segment \( d\mathbf{L} \) of the wire can be expressed using the formula:
\[
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