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Question: A wire carrying a current of 4A is in a 0.5T magnetic field as shown above. If the direction of elec...

A wire carrying a current of 4A is in a 0.5T magnetic field as shown above. If the direction of electron flow is to the right as shown, what is the magnitude and direction of the force (per unit length) on the wire?

A. 2N/m into the page
B. 8N/m out of the page
C. 2N/m to the top of the page
D. 2N/m to the bottom of the page
E. 8N/m to the top of the page

Explanation

Solution

From the question given you could simply jolt down the details. Since this question specifically asks for the direction, one should be careful while noting that down. Then you could find the magnetic force from the expression in terms of the given quantities and then divide it by the length to get the answer.
Formula used:
Magnetic force,
F→=I(l→×B→)\overrightarrow{F}=I\left( \overrightarrow{l}\times \overrightarrow{B} \right)

Complete step by step answer:
In the question, we are given a wire carrying 4A current in the presence of a 0.5T magnetic field. We are given a figure depicting this situation and we are supposed to find the magnitude and direction of the force per unit length in the wire.

In the above figure, we see that, the magnetic field with direction can be given by,
B→=−0.5z^\overrightarrow{B}=-0.5\widehat{z} ……………………………….. (1)
Now, we have the current flow opposite to that of the electron flow given in the figure,
I→=−4Ax^\overrightarrow{I}=-4A\widehat{x} …………………………………… (2)
For l length of the wire, we have the magnetic force on it given by,
F→=I(l→×B→)\overrightarrow{F}=I\left( \overrightarrow{l}\times \overrightarrow{B} \right)
From (1) and (2), we have,
F→=IlB(−x^×(−z^))=IlB(−y^)\overrightarrow{F}=IlB\left( -\widehat{x}\times \left( -\widehat{z} \right) \right)=IlB\left( -\widehat{y} \right)
So, the force per unit length would be,
F→l=(4×0.5)(−y^)=−2y^Nm−1\dfrac{\overrightarrow{F}}{l}=\left( 4\times 0.5 \right)\left( -\widehat{y} \right)=-2\widehat{y}N{{m}^{-1}}
So, the correct answer is “Option D”.

Note: It is a basic knowledge that cross for magnetic field means it is directed into the paper and dot means it is directed outwards a paper. So for the coordinate system taken as reference in the figure we see that the magnetic field is directed into the paper as it is marked as cross in the figure.