A point charge of [tex][tex]$5.7 \mu C$[/tex][/tex] moves at [tex][tex]$4.5 \times 10^5 \, m/s$[/tex][/tex] in a magnetic field that has a field strength of [tex][tex]$3.2 \, mT$[/tex][/tex], as shown in the diagram.

What is the magnitude of the magnetic force acting on the charge?

A. [tex][tex]$6.6 \times 10^{-3} \, N$[/tex][/tex]
B. [tex][tex]$4.9 \times 10^{-3} \, N$[/tex][/tex]
C. [tex][tex]$4.9 \times 10^3 \, N$[/tex][/tex]
D. [tex][tex]$6.6 \times 10^3 \, N$[/tex][/tex]



Answer :

To solve for the magnitude of the magnetic force acting on a moving charge in a magnetic field, we use the formula:

[tex]\[ F = q \cdot v \cdot B \cdot \sin(\theta) \][/tex]

where:
- [tex]\( F \)[/tex] is the magnetic force,
- [tex]\( q \)[/tex] is the charge,
- [tex]\( v \)[/tex] is the velocity of the charge,
- [tex]\( B \)[/tex] is the magnetic field strength,
- [tex]\(\theta\)[/tex] is the angle between the velocity and the magnetic field direction.

Given the values:
- [tex]\( q = 5.7 \mu C = 5.7 \times 10^{-6} \, C \)[/tex],
- [tex]\( v = 4.5 \times 10^5 \, m/s \)[/tex],
- [tex]\( B = 3.2 \, mT = 3.2 \times 10^{-3} \, T \)[/tex],
- [tex]\( \theta = 90^\circ \)[/tex] (since this angle will maximize the force, and [tex]\(\sin(90^\circ) = 1\)[/tex]),

we can plug these values into the formula. Since the angle [tex]\(\theta\)[/tex] is [tex]\(90^\circ\)[/tex]:

[tex]\[ \sin(90^\circ) = 1 \][/tex]

Now we compute:

[tex]\[ F = (5.7 \times 10^{-6} \, C) \cdot (4.5 \times 10^5 \, m/s) \cdot (3.2 \times 10^{-3} \, T) \cdot 1 \][/tex]

Let's multiply the values step-by-step:

1. Calculate [tex]\( q \cdot v \)[/tex]:

[tex]\[ 5.7 \times 10^{-6} \, C \cdot 4.5 \times 10^5 \, m/s = 2.565 \times 10^{-6+5} = 2.565 \times 10^{-1} \][/tex]

2. Multiply the result by [tex]\( B \)[/tex]:

[tex]\[ 2.565 \times 10^{-1} \cdot 3.2 \times 10^{-3} = 8.208 \times 10^{-4} \][/tex]

So, the magnitude of the magnetic force is:

[tex]\[ F = 0.008208 \, N \][/tex]

Hence, the correct magnitude of the magnetic force acting on the charge is [tex]\( 0.008208 \, N \)[/tex].

Among the choices given:
[tex]\( 6.6 \times 10^{-3} \, N \)[/tex],
[tex]\( 4.9 \times 10^{-3} \, N \)[/tex],
[tex]\( 4.9 \times 10^3 \, N \)[/tex],
[tex]\( 6.6 \times 10^3 \, N \)[/tex],

the value closest to our computed result of [tex]\( 0.008208 \, N \)[/tex] is:

[tex]\[ 6.6 \times 10^{-3} \, N \][/tex]

Therefore, the correct answer is:
[tex]\[ \boxed{6.6 \times 10^{-3} \, N} \][/tex]