Question

In which direction does a magnetic force act on an electron that is moving in the positive \(x\) -direction in a magnetic field pointing in the positive \(z\) -direction? a) the positive \(y\) -direction b) the negative \(y\) -direction c) the negative \(x\) -direction d) any direction in the \(x y\) -plane

Short answer

Answer: b) the negative y-direction. Explanation: Using the Lorentz force equation and the right-hand rule, the magnetic force acts in the positive y-direction for a positive charge, but since the electron has a negative charge, the force is in the opposite direction, i.e., the negative y-direction.

Step-by-step solution

Determine the electron's charge

An electron has a negative charge, which can be represented as \(q = -e\), where \(e\) is the elementary charge (\(1.6 \times 10^{-19}\) Coulombs).

Write down the known quantities/vector components

An electron is moving in the positive \(x\)-direction, so its velocity vector is \(\vec{v} = v_x\hat{i}\), where \(v_x > 0\). The magnetic field is in the positive \(z\)-direction, so the magnetic field vector is \(\vec{B} = B_z\hat{k}\), where \(B_z > 0\).

Compute the cross product

The magnetic force acting on the electron is given by the Lorentz force equation: \[ \vec{F} = q(\vec{v} \times \vec{B}) \] Since \(q=-e\), we have: \[ \vec{F} = -e(\vec{v} \times \vec{B}) \] To take the cross product, we can use the determinant formula: \[ \vec{F} = -e\begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ v_x & 0 & 0 \\ 0 & 0 & B_z \end{vmatrix} \] Applying the determinant formula, we find: \[ \vec{F} = -e(\hat{i}(0 - 0) - \hat{j}(0 - 0) + \hat{k}(0 - 0)) = -e\cdot 0 \] So, the magnetic force is zero in the \(x\) and \(z\) directions.

Find the direction of the magnetic force in the \(y\)-direction

Now we need to use the right-hand rule to find the direction of the magnetic force in the \(y\)-direction. Point your right-hand fingers in the direction of the electron's velocity (\(+x\)), and then curl them toward the direction of the magnetic field (\(+z\)). Your thumb will be pointing in the direction of the force. Since the electron has a negative charge, the actual force will be in the opposite direction. In this case, your thumb will be pointing in the positive \(y\)-direction, but since the electron has a negative charge, the force is in the negative \(y\)-direction. Therefore, the correct answer is: b) the negative \(y\)-direction