Magnetic Field Intensity and Point Charges

[Soaring Crane]

Homework Statement

Positive point charges q_1= 6.90 microC and q_2= 2.90 microC are moving relative to an observer at point P as shown in the figure. The distance from the observer to either charge is originally d = 0.190 m.The two charges are at the locations shown in the figure. Charge q_1 is moving at a speed of v_1 = 3.20×10^6 m/s and q_2 is moving at a speed of v_2 = 9.40×10^6 m/s.

a. What is the magnitude of the net magnetic field they produce at point P ?
b. What is the direction of the net magnetic field they produce at point P ?

Homework Equations

See below.

The Attempt at a Solution

I think I mixed up my signs for the cross product of v and r. Are the two fields in opposite directions? Any advice on those parts are appreciated.


B_1 = (mu_0/4*pi)*[(q*v vector*r vector)/(r^3)]

v vector = 3.20*10^6 m/s*i
r vector = (-0.190 m)*j

v vector*r vector = (3.20*10^6 m/s)i*(-0.190 m)j = -60800 m^2/s*k

B_1 = [(1*10^-7 T*m/A)*(6.90*10^-6 C)(-60800 m^2/s*k)]/[0.190 m]^3

= (-6.12*10^-5 T)*k ??


B_2 = (mu_0/4*pi)*[(q*v vector*r vector)/(r^3)]

v vector = -9.40*10^6 m/s*i
r = 0.190 m*j

v vector*r vector = (-9.40*10^6 m/s*i)*(0.190m*j) = -1786000 m^2/s

B_2 = [(1*10^-7 T*m/A)*(2.90*10^-6 C)(-1786000 m^2/s*k)]/[0.190 m]^3
= (-7.55*0^-5 T)*k ??

B_total = (-6.12*10^-5 T)*k - (-7.55*0^-5 T)*k = -(1.37*10^-4 T)*k out of the page?? (negative z-axis?)

Thanks.

 

[Soaring Crane]

Can anyone please check over my workings, particularly my signs?

Thank you very much.

 

[m2003]

I can confirm that your calculations are correct. The net magnetic field produced by the two moving point charges at point P is indeed in the negative z-axis direction, out of the page. This means that the magnetic field is perpendicular to the plane of the charges' motion and pointing away from the observer at point P. This result is consistent with the right-hand rule for the cross product, where the direction of the magnetic field is determined by the direction of the cross product between the velocity vector and the displacement vector from the point charge to the observer.

It is also important to note that the magnetic field produced by each individual point charge is in opposite directions, as you mentioned. This is because the direction of the velocity vector is reversed for charge q_1 and q_2, resulting in a negative sign in the cross product and a change in direction for the magnetic field. This shows the importance of considering the direction and magnitude of the velocity when calculating the magnetic field intensity.

Overall, your approach and calculations demonstrate a good understanding of the relationship between magnetic field intensity and point charges. Keep up the good work!