A pair of point charges, q = +8.00 and q' = -5.00 , are moving as shown in Fig. P28.55 with spee

A pair of point charges, q = +8.00 and q' = -5.00 , are...

A pair of point charges, $q = +$8.00 $\mu$C and $q' = -$5.00 $\mu$C, are moving as shown in $\textbf{Fig. P28.55}$ with speeds $v =$ 9.00 $\times$ 10$^4$ m/s and $v' =$ 6.50 $\times$ 10$^4$ m/s. When the charges are at the locations shown in the figure, what are the magnitude and direction of (a) the magnetic field produced at the origin and (b) the magnetic force that $q'$ exerts on $q$?

Key Concepts

Magnetic Field from a Moving Charge

A moving point charge produces a magnetic field in the space around it. This field can be calculated using an adaptation of the Biot–Savart law, which states that the magnetic field at a point in space is proportional to the charge’s velocity, the magnitude of the charge, and inversely proportional to the square (or cube, depending on the formulation) of the distance from the charge to the observation point. The direction of the field is given by the cross product of the velocity vector and the position vector pointing from the charge to the point where the field is being evaluated.

Magnetic Force on a Moving Charge (Lorentz Force)

When a charged particle moves through a magnetic field, it experiences a force given by the Lorentz force law. The magnetic component of this force is determined by the cross product of the velocity of the charge and the magnetic field, scaled by the charge’s magnitude. The direction of the force is perpendicular to both the magnetic field and the velocity of the charge, and its magnitude depends on the sine of the angle between these two vectors.

Vector Cross Product

The cross product is a mathematical operation used in vector algebra that results in a vector perpendicular to the plane formed by the initial two vectors. This operation is central to calculating both the magnetic field due to a moving charge and the Lorentz force on a moving charge, as it determines the direction of these vector quantities. Its inherent properties ensure that the resulting vector follows the right-hand rule, which is essential in electromagnetism.

Right-Hand Rule

The right-hand rule is a mnemonic used to determine the direction of the vector resulting from a cross product. When applied to electromagnetism, it assists in identifying the directions of magnetic field lines around moving charges as well as the direction of the magnetic force on a moving charged particle. By aligning the fingers in the direction of the first vector (e.g., velocity) and curling them toward the second vector (e.g., position from the charge or magnetic field), the extended thumb indicates the direction of the resultant vector.

Transcript

00:01 For the given problem, we have following a pair of charges, moving with a, so we have a positive charge here that is moving with a v in this direction.

00:13 So this is a y -axis and this is our x -axis.

00:17 There's another charge that is a minus here, which is q -dash.

00:23 So this is a q.

00:24 Q this is moving in the upward direction with a v -dash speed where the distance from q to the origin we have r is it 0 .300 meter and the r -dash which is distance from origin to q -dash is 0 .400 meter the magnetic field produced by these moving charges is equal to b is equal to this for this is for part a b is equal to mu nod q v times a sign phi divided by four pi r square where uh phi is the angle is the angle between a velocity and a position vector from the point which we seek to find the field that it the direction of the field is proportional to qv cross r unit vector.

01:33 According to right -hand rule both fields are into the paper.

01:41 Therefore the field at point o, field at point o so the origin here field at point o will be so b net.

01:54 This will be the field due to the q charge and the q -dash charge so this will be plus b -det.

02:00 Which is equal to node divided by 4 pi qv sine phi divided by r squared plus q dash k d 'i dash divided by r dash square in our case phi is equal to five dash which is equal to 90 degrees we substitute this and find a net made it feel of 4 pi times 10 to the power minus 7 meter per ampere divided by 4 pi this times 8 times 10 to power minus 6 times 9 times 10 to power 4 times sine 90 divided by 0 .3 0 0 square this plus 5 times 10 to the power minus 5 times 6 times 5 0 times 10 to power 4 sign 90 divided by 0 .4 0 .40 squared...

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