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A -4.80@mC charge is moving at a constant speed of 6.80 * 105 m>s in the +x-direction relative to a reference frame. At the instant when the point charge is at the origin, what is the magnetic-field vector it produces at the following points:

(a) x= 0.500 m, y= 0, z= 0;

(b) x= 0, y= 0.500 m, z= 0;

(c) x= 0.500 m, y= 0.500 m, z= 0;

(d) x= 0, y= 0, z= 0.500 m?

Short Answer

Expert verified

a)B=0b)B=-1.31×10-6Tk^c)B=-4.62×10-7Tk^d)B=1.31×10-6Tk^

Step by step solution

01

Solve part (a) of the problem.

Consider a charge (q=-4.80) which is moving at a constant speed ofv=6.80×105m/s in the +x direction relative to a reference frame. The magnetic field produced by the moving charge is given by,

B=μo4πqv×r^r2 (1)

We need to find the magnetic field vector that produces at the point = 0.500 m, y =0, z = 0 at the instant when the point charge is at the origin. In this ca have v=vi^,r=ri^so,

v×r=0

thus,

B=0

02

Solve part (b) of the problem.

Now we need to find the magnetic field vector that produces at the point z = 0,y = 0.500 m, and z =0 at the instant when the point charge is at the origin. In this v=vi^,r=rj^so,

v×r=vk^

where r=x2+y2+z2=y=0.500Substitute into (1) we get,

B=μ04πqvr2k^=1.0×107Ns2/C24.80×106C6.80×105m/s(0.500m)2k^=1.31×106Tk^B=1.31×106Tk^

03

Solve part (c) of the problem.

Now we need to find the magnetic field vector that produces at the point z= 0.500 m, y = 0.500 m, and z = 0 at the instant when the point charge is at the origin. In this case, we have v=vi^,r=(0.500)(i^+j^)so,

role="math" localid="1668229000516" v×r=(0.500)vk^

where m r=x2+y2+z2=x2+y2=0.7071mSubstitute into (1) we get

B=μ04πq(0.500)vr3k^=1.0×107Ns2C24.80×106C(0.500)6.80×105m/s(0.500m)3k^=4.62×107Tk^B=4.62×107Tk^

04

Solve part (d) of the problem.

Finally, we need to find the magnetic field vector that produces at the point r = 0, y =0, z =0.500 m at the instant when the point charge is at the origin. In this case, we have v=vi^,r=rk^ so,

v×r=vk^

thus,

B=μ04πq(0.500)vr2k^=1.0×107Ns2/C24.80×106C6.80×105m/s(0.500m)2k^=1.31×106Tk^B=1.31×106Tk^

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