Question

An electron that is moving through a uniform magnetic field has velocity $\overrightarrow{\mathbf{v}}\mathbf{=}\mathbf{(}\mathbf{40}\mathbf{}\mathbf{\text{km/s}}\mathbf{)}\mathbf{+}\mathbf{(}\mathbf{35}\mathbf{}\mathbf{\text{km/s}}\mathbf{)}$when it experiences a force $\overrightarrow{\mathbf{F}}\mathbf{=}\mathbf{-}\mathbf{(}\mathbf{4}\mathbf{.2}\mathbf{\text{\hspace{0.17em}fN}}\mathbf{)}\mathbf{+}\mathbf{(}\mathbf{4}\mathbf{.8}\mathbf{\text{\hspace{0.17em}fN}}\mathbf{)}$due to the magnetic field. If ${\mathit{B}}_{\mathbf{x}}\mathbf{=}\mathbf{0}$, calculate the magnetic field B.

Short answer

Magnetic field is equal to $\overrightarrow{B}=\left(0.75\text{T}\right)$

Step-by-step solution

Identification of given data

$\begin{array}{l}\overrightarrow{v}=(40\text{km/s})+(35\text{km/s})\\ \overrightarrow{F}=-\left(4.2\text{\hspace{0.17em}fN}\right)+\left(4.8\text{\hspace{0.17em}fN}\right)\\ B_x=0\end{array}$

Significance of magnetic force

The influence of a magnetic field produced by one charge on another is what is known as the magnetic force between two moving charges

We can use the formula for force on the charged particle in the magnetic field. Writing this formula in the vector form and comparing it with the given information, we can find the magnetic field.

Formula:

$\overrightarrow{F}=q\left(\overrightarrow{v}\times \overrightarrow{B}\right)$

Determining the magnetic field B→

We are given that $B_x=0$, the velocity in the z direction, and the magnetic force in the z direction is zero.

We have

$\overrightarrow{F}=q(\overrightarrow{v}\times \overrightarrow{B})$

Using cross product formula in vector notation and using the above given terms, we can write

$F_x\hat{i}+F_y\hat{j}=q\left(v_y{B}_z\hat{i}-v_x{B}_z\hat{j}+v_x{B}_y\hat{k}\right)$

From the above equation, we can see that $B_y$is zero.

Now $B_z$can be found from the above equation and force vector.

$\begin{array}{c}{B}_z=\frac{F_x}{q{v}_y}\\ =\frac{-4.2\text{fN}}{-1.6\times {10}^{-19}\text{C}\times 35\text{km/s}}\end{array}$

Writing the terms in the SI units, we have

$\begin{array}{c}{B}_z=\frac{4.2\times {10}^{-15}\text{N}}{1.6\times {10}^{-19}\text{\hspace{0.17em}C}\times 35,000\text{m/s}}\\ =0.75\text{T}\end{array}$

Therefore, magnetic field in vector notation can be written as

$\overrightarrow{B}=(0.75\text{T})$