Question

An electron moves at 0.130 c as shown in the figure (Figure 1). There are points: A, B, C, and D 2.10 μm from the electron.

a) Find the magnitude of the magnetic field this electron produces at the point A .
b) Find the magnitude of the magnetic field this electron produces at the point B .
c) Find the magnitude of the magnetic field this electron produces at the point C .
d) Find the magnitude of the magnetic field this electron produces at the point D

Answer 1

Hi there!

We can use Biot-Savart's Law for a moving particle:

`B= (\mu_0 )/(4\pi)\frac{q\vec{v}* \vec{r}}{r^2 }`

B = Magnetic field strength (T)
v = velocity of electron (0.130c = 3.9 × 10⁷ m/s)

q = charge of particle (1.6 × 10⁻¹⁹ C)

μ₀ = Permeability of free space (4π × 10⁻⁷ Tm/A)

r = distance from particle (2.10 μm)

There is a cross product between the velocity vector and the radius vector (not a quantity, but specifies a direction). We can write this as:

`B= (\mu_0 )/(4\pi)\frac{q\vec{v} \vec{r}sin\theta}{r^2 }`

Where 'θ' is the angle between the velocity and radius vectors.

a)
To find the angle between the velocity and radius vector, we find the complementary angle:

θ = 90° - 60° = 30°

Plugging 'θ' into the equation along with our other values:

`B= (\mu_0 )/(4\pi)\frac{q\vec{v} \vec{r}sin\theta}{r^2 }\\\\B= ((4\pi *10^(-7)))/(4\pi)\frac{(1.6*10^(-19))(3.9*10^(7)) \vec{r}sin(30)}{(2.1*10^(-5))^2 }`

`B = \boxed{7.07 *10^(-10) T}`

b)
Repeat the same process. The angle between the velocity and radius vector is 150°, and its sine value is the same as that of sin(30°). So, the particle's produced field will be the same as that of part A.

c)

In this instance, the radius vector and the velocity vector are perpendicular so

'θ' = 90°.

`B= ((4\pi *10^(-7)))/(4\pi)\frac{(1.6*10^(-19))(3.9*10^(7)) \vec{r}sin(90)}{(2.1*10^(-5))^2 } = \boxed{1.415 * 10^(-9)T}`

d)
This point is ALONG the velocity vector, so there is no magnetic field produced at this point.

Aka, the radius and velocity vectors are parallel, and since sin(0) = 0, there is no magnetic field at this point.

`\boxed{B = 0 T}`