An electron travels with v⃗=(5.0×106 i^) m/s\vec{v} = (5.0 \times $$10^6$$\,\hat{i})\,\text{m/s} through a point in space where E⃗=(2.0×105 i^−2.0×105 j^) V/m\vec{E} = (2.0 \times $$10^5$$\,\hat{i} - 2.0 \times $$10^5$$\,\hat{j})\,\text{V/m} and B⃗=−0.10 k^ T\vec{B} = -0.10\,\hat{k}\,\text{T}. What is the force on the electron?

Ch 31: Electromagnetic Fields and Waves

Knight Calc - Physics for Scientists and Engineers 5th Edition

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Knight Calc 5th Edition

Ch 31: Electromagnetic Fields and Waves

Problem 31

Chapter 31, Problem 31

An electron travels with $\vec{v} = (5.0 \times 10^{6} \hat{i}) m/s$ through a point in space where $\vec{E} = (2.0 \times 10^{5} \hat{i} - 2.0 \times 10^{5} \hat{j}) V/m$ and $\vec{B} = - 0.10 \hat{k} T$ . What is the force on the electron?

Verified step by step guidance

Step 1: Recall the formula for the force on a charged particle in the presence of both electric and magnetic fields. This is given by the Lorentz force equation:

F⃗ = q(E⃗ + v⃗ × B⃗)

, where

q

is the charge of the particle,

E⃗

is the electric field,

v⃗

is the velocity, and

B⃗

is the magnetic field.

Step 2: Substitute the known values into the equation. The charge of an electron is

q = -1.6 × 10^{-19}\(\text{ C}\)

. The given values are

v⃗ = (5.0 × 10^6 \, \(\text{m/s}\)) \, \(\text{in the }\) \, \(\text{i}\) \, \(\text{direction}\)

E⃗ = (2.0 × 10^5 \, \(\text{V/m}\)) \, \(\text{in the }\) \, \(\text{i}\) \, \(\text{direction}\) - (2.0 × 10^5 \, \(\text{V/m}\)) \, \(\text{in the }\) \, \(\text{j}\) \, \(\text{direction}\)

, and

B⃗ = -0.10 \, \(\text{T}\) \, \(\text{in the }\) \, \(\text{k}\) \, \(\text{direction}\)

Step 3: Calculate the cross product

v⃗ × B⃗

. Using the determinant method for cross products,

v⃗ × B⃗ = \(\begin{vmatrix}\) \(\hat{i}\) & \(\hat{j}\) & \(\hat{k}\) \\ 5.0 × 10^6 & 0 & 0 \\ 0 & 0 & -0.10 \(\end{vmatrix}\)

. Expand this determinant to find the components of the cross product.

Step 4: Add the electric field contribution and the magnetic field contribution to find the total force. The electric field contribution is

qE⃗

, and the magnetic field contribution is

q(v⃗ × B⃗)

. Combine these two vector components to get the total force vector.

Step 5: Multiply the resulting vector by the charge of the electron,

q = -1.6 × 10^{-19}\(\text{ C}\)

, to account for the sign and magnitude of the electron's charge. This will give the final force vector acting on the electron.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Lorentz Force

The Lorentz force is the force experienced by a charged particle moving through electric and magnetic fields. It is given by the equation F = q(E + v × B), where F is the force, q is the charge of the particle, E is the electric field, v is the velocity of the particle, and B is the magnetic field. This concept is crucial for understanding how charged particles like electrons interact with electromagnetic fields.

Electric Field (E)

An electric field is a region around a charged particle where other charged particles experience a force. It is represented by the vector E, measured in volts per meter (V/m). The direction of the electric field is defined as the direction a positive test charge would move. In this question, the electric field influences the force acting on the electron as it moves through space.

Magnetic Field (B)

A magnetic field is a vector field that exerts a force on moving charges and magnetic dipoles. It is represented by the vector B, measured in teslas (T). The magnetic force on a charged particle is perpendicular to both the velocity of the particle and the magnetic field direction, which can lead to circular or helical motion. Understanding the magnetic field is essential for calculating the total force on the electron in this scenario.

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