Question

A wire of arbitrary shape carries a current \[I = 2{\text{A}}\], consider the portion of wire between $(0,0,0)$ and $(4m,4m,4m)$. A magnetic field is given by $B = 1.2 \times {10^{ - 4}}T\widehat i + 2.0 \times {10^{ - 4}}T\widehat j$ exists in the space. The force acting on the given portion is:

A. Incalculable as the length of wire is not known
B. $\overrightarrow F = [(\widehat i + \widehat j + \widehat k) \times (1.2\widehat i + 2.0\widehat j)]{\text{N}}$
C. $\overrightarrow F = 8 \times {10^{ - 4}}[(\widehat i + \widehat j + \widehat k) \times (1.2\widehat i + 2.0\widehat j)]{\text{N}}$
D. $\overrightarrow F = 8 \times {10^{ - 4}}[(1.2\widehat i + 2.0\widehat j) \times (\widehat i + \widehat j + \widehat k)]{\text{N}}$

Answer 1

Hint: We are given the values of the current and magnetic field and we need to find the force acting on the current-carrying conductor present in a magnetic field.We can find the value of force using the relation between force and current and magnetic field and we can find the direction of force using the right-hand thumb rule.

Complete step by step answer:
Given, \[I = 2{\text{A}}\] and $B = 1.2 \times {10^{ - 4}}T\widehat i + 2.0 \times {10^{ - 4}}T\widehat j$.
We will the following formula
$\overrightarrow F = I(\overrightarrow l \times \overrightarrow B )$
Where $\overrightarrow l $ is the shortest distance between two points on which the force is expected to be measured and the direction is in the direction of the current.
$\overrightarrow l = (4 - 0)\widehat i + (4 - 0)\widehat j + (4 - 0)\widehat k$
$ \Rightarrow \overrightarrow l = 4\widehat i + 4\widehat j + 4\widehat k$
Now substituting the values in the force formula we get
$ \Rightarrow \overrightarrow F = 2[4\widehat i + 4\widehat j + 4\widehat k] \times [1.2 \times {10^{ - 4}}T\widehat i + 2.0 \times {10^{ - 4}}T\widehat j]$
$\therefore \overrightarrow F = 8 \times {10^{ - 4}}[(\widehat i + \widehat j + \widehat k) \times (1.2\widehat i + 2.0\widehat j)]{\text{N}}$

Therefore the correct option is C.

Additional information: A magnetic field is a vector quantity that describes the magnetic influence on moving electric charges, electric currents and magnetic materials. A charge moving in a magnetic field experiences a force normal to its velocity and the magnetic field which is known as the Lorentz force. Lorentz force is the force exerted on a charged particle moving with a velocity through an electric field and a magnetic field. The entire electromagnetic force on the charged particle is known as the Lorentz force.

Note: All the quantities involved in the problems are vector quantities and does we used the vector law of multiplication for finding the product and we used the displacement between the two points given in the question which is the shortest distance between them and thus we have denoted it as a vector quantity.