A positive point charge moves with constant velocity along the $x$-axis. At a point located on the positive $z$-axis, what is the direction of the magnetic field produced by the moving charge at that point?

• A. In the negative $y$-direction
• B. In the positive $y$-direction
• C. In the positive $z$-direction
• D. In the positive $x$-direction

Answer: (A)

Verified step by step guidance

1. Identify the velocity vector \( \vec{v} \) of the positive charge. Since the charge moves along the x-axis with constant velocity, we write \( \vec{v} = v \hat{x} \), where \( v \) is the speed and \( \hat{x} \) is the unit vector in the x-direction.
2. Determine the position vector \( \vec{r} \) from the charge to the point on the positive z-axis. Since the point lies on the z-axis, \( \vec{r} = z \hat{z} \), where \( z > 0 \) and \( \hat{z} \) is the unit vector in the z-direction.
3. Recall the formula for the magnetic field \( \vec{B} \) produced by a moving point charge: \[ \vec{B} = \frac{\mu_0}{4\pi} \frac{q \vec{v} \times \hat{r}}{r^2} \] where \( q \) is the charge, \( \vec{v} \) is the velocity, \( \hat{r} = \frac{\vec{r}}{r} \) is the unit vector from the charge to the point, and \( r = |\vec{r}| \) is the distance.
4. Calculate the cross product \( \vec{v} \times \hat{r} \). Since \( \vec{v} = v \hat{x} \) and \( \hat{r} = \hat{z} \), use the right-hand rule or the standard vector cross product: \[ \hat{x} \times \hat{z} = -\hat{y} \] Therefore, \( \vec{v} \times \hat{r} = v (-\hat{y}) = -v \hat{y} \).
5. Conclude that the magnetic field \( \vec{B} \) at the point on the positive z-axis points in the negative y-direction, because the magnetic field direction is given by the direction of \( \vec{v} \times \hat{r} \).