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Chapter 15: Problem 17

Gegeben sind drei Punktladungen \(q_{0}=-4 C\), \(q_{1}=6 C\) und \(q_{2}=3 C\) im \(\mathbb{R}^{2}\) an den jeweiligen Stellen \(\boldsymbol{r}_{0}=\left(\begin{array}{l}0 \\ 0\end{array}\right), \boldsymbol{r}_{2}=\left(\begin{array}{l}1 \\ 0\end{array}\right)\) und \(\boldsymbol{r}_{3}=\left(\begin{array}{l}1 \\ 1\end{array}\right)\) Bestimmen Sie die resultierende Kraft \(\boldsymbol{F}\), die von \(q_{1}\) und \(q_{2}\) auf \(q_{0}\) ausge眉bt wird.

Short Answer

Expert verified
Answer: The net force exerted on q_0 by q_1 and q_2 is F = (-215.76, -71.92) N.

Step by step solution

01

Calculate the distance between charges

For each pair of charges (\(q_0\) and \(q_1\), and \(q_0\) and \(q_2\)), we need to calculate the distance between them. This can be done using the given positions: Distance between \(q_0\) and \(q_1\): \(\boldsymbol{r}_{10} = \boldsymbol{r}_{1} - \boldsymbol{r}_{0} = \left(\begin{array}{l}1 \\\ 0\end{array}\right) - \left(\begin{array}{l}0 \\\ 0\end{array}\right) = \left(\begin{array}{l}1 \\\ 0\end{array}\right)\) Distance between \(q_0\) and \(q_2\): \(\boldsymbol{r}_{20} = \boldsymbol{r}_{2} - \boldsymbol{r}_{0} = \left(\begin{array}{l}1 \\\ 1\end{array}\right) - \left(\begin{array}{l}0 \\\ 0\end{array}\right) = \left(\begin{array}{l}1 \\\ 1\end{array}\right)\)
02

Calculate the force exerted by each charge on \(q_0\)

Using Coulomb's Law, we can calculate the force exerted by each charge on \(q_0\). The formula for calculating the force between two charges is given by: \(\boldsymbol{F} = k\frac{q_{1}q_{2}}{r^{2}}\boldsymbol{\hat{r}}\) where \(k\) is Coulomb's constant (\(8.99 \times 10^{9} Nm^{2}C^{-2}\)), \(q_1\) and \(q_2\) are the magnitudes of the charges, \(r\) is the distance between the charges, and \(\boldsymbol{\hat{r}}\) is the unit vector pointing from the source charge to the target charge. Force exerted by \(q_1\) on \(q_0\): \(\boldsymbol{F}_{10} = k\frac{q_0q_1}{r_{10}^2}\boldsymbol{\hat{r}}_{10} = 8.99\times10^9 \times \frac{-4\times6}{1^2}\left(\begin{array}{l}1 \\\ 0\end{array}\right) = -143.84 \times \left(\begin{array}{l}1 \\\ 0\end{array}\right)N\) Force exerted by \(q_2\) on \(q_0\): \(\boldsymbol{F}_{20} = k\frac{q_0q_2}{r_{20}^2}\boldsymbol{\hat{r}}_{20} = 8.99\times10^9 \times \frac{-4\times3}{(\sqrt{2})^2}\left(\frac{\left(\begin{array}{l}1 \\\ 1\end{array}\right)}{\sqrt{2}}\right) = -71.92 \times \left(\begin{array}{l}1 \\\ 1\end{array}\right)N\)
03

Calculate the net force on \(q_0\)

Now that we have the forces exerted by each charge on \(q_0\), we simply add the force vectors together to obtain the net force: \(\boldsymbol{F}_{net} = \boldsymbol{F}_{10} + \boldsymbol{F}_{20} = \left(\begin{array}{l}-143.84 \\\ 0\end{array}\right)N + \left(\begin{array}{l}-71.92 \\\ -71.92\end{array}\right)N = \left(\begin{array}{l}-215.76 \\\ -71.92\end{array}\right)N\) The net force exerted on \(q_0\) by \(q_1\) and \(q_2\) is \(\boldsymbol{F} = \left(\begin{array}{l}-215.76 \\\ -71.92\end{array}\right)N\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Electric Force
Coulomb's Law describes the electric force between two point charges. This force is essential for understanding how charged particles interact.

The electric force between two charges can be either attractive or repulsive.
  • Attractive Force: Occurs between charges with opposite signs.
  • Repulsive Force: Occurs between charges with the same sign.

The electric force is calculated using Coulomb's Law, given by:
\[ \boldsymbol{F} = k \frac{q_1 q_2}{r^2} \boldsymbol{\hat{r}} \]where
  • \(k\) is the Coulomb's constant \(8.99 \times 10^{9} \, Nm^{2}C^{-2}\),
  • \(q_1\) and \(q_2\) are the magnitudes of the charges,
  • \(r\) is the distance between the charges,
  • \(\boldsymbol{\hat{r}}\) is the unit vector from the source charge to the target charge.

Understanding these details allows us to determine the nature and strength of interactions between charged particles.
Vector Addition
When calculating the net force on a charge, it is vital to consider all forces acting on it. These forces usually come from other charges within the vicinity. In order to obtain the resultant force, we use a process called vector addition.

Vector addition involves:
  • Adding component-wise values. This means adding all x-components together and all y-components together.
  • Using the vector representation \(\left( \begin{array}{c} x \ y \end{array} \right)\), you sum up each part separately.
  • Visualizing forces graphically can help to understand how they might cancel each other out or combine.

For example, if two vectors \(\boldsymbol{F}_1 = \left( \begin{array}{c} a \ b \end{array} \right)\) and \(\boldsymbol{F}_2 = \left( \begin{array}{c} c \ d \end{array} \right)\) act on a point, the net force \(\boldsymbol{F}_{net}\) is given by:
\[ \boldsymbol{F}_{net} = \left( \begin{array}{c} a+c \ b+d \end{array} \right) \]This approach simplifies problems involving multiple forces acting at angles.
Point Charges
Point charges can be considered as charges concentrated at a single point in space. This concept is widely used in physics to simplify the study of electric forces and fields.

Key properties of point charges:
  • They have no size, only a magnitude of charge.
  • Real-life representations can include objects like charged spheres where most of the charge is effectively at the center.
  • Using point charges makes calculating forces simpler since Coulomb's law directly applies to them.

These charges can be positive or negative, influencing whether the force between them is attractive or repulsive. Understanding point charges is foundational for many electrodynamics problems where complex bodies are reduced to simpler equivalent point charges.
Distance Calculation
Calculating the distance between charges is a fundamental step in determining the force between them using Coulomb鈥檚 Law.

To find this distance:
  • Use the position coordinates of each charge in the problem space, typically represented as vectors.
  • Apply the formula for the distance between two points:
    \[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] where \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two charges.
  • This formula arises from the Pythagorean theorem which relates distances in the Cartesian plane.

In practice, use these distances to determine the force exerted on a point charge by using them in the expression for Coulomb's Law. This ensures precise and accurate calculation of forces in physical systems.

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Most popular questions from this chapter

Gelten in einem Vektorraum \(V\) die folgenden Aussagen? (a) Ist eine Basis von \(V\) unendlich, so sind alle Basen von \(V\) unendlich. (b) Ist eine Basis von \(V\) endlich, so sind alle Basen von \(V\) endlich. (c) Hat \(V\) ein unendliches Erzeugendensystem, so sind alle Basen von \(V\) unendlich. (d) Ist eine linear unabh盲ngige Menge von \(V\) endlich, so ist es jede.

Gibt es f眉r jede nat眉rliche Zahl \(n\) eine Menge \(A\) mit \(n+1\) verschiedenen Vektoren \(\boldsymbol{v}_{1}, \ldots, \boldsymbol{v}_{n+1} \in \mathbb{R}^{n}\), sodass je \(n\) Elemente von \(A\) linear unabh盲ngig sind? Geben Sie eventuell f眉r ein festes \(n\) eine solche an.

Geben Sie zu folgenden Teilmengen des \(\mathbb{R}\)-Vektorraums \(\mathbb{R}^{3}\) an, ob sie Untervektorr盲ume sind, und begr眉nden Sie dies: (a) \(U_{1}:=\left\\{\left(\begin{array}{l}v_{1} \\ v_{2} \\\ v_{3}\end{array}\right) \in \mathbb{R}^{3} \mid v_{1}+v_{2}=2\right\\}\) (b) \(U_{2}:=\left\\{\left(\begin{array}{l}v_{1} \\ v_{2} \\\ v_{3}\end{array}\right) \in \mathbb{R}^{3} \mid v_{1}+v_{2}=v_{3}\right\\}\) (c) \(U_{3}:=\left\\{\left(\begin{array}{l}v_{1} \\ v_{2} \\\ v_{3}\end{array}\right) \in \mathbb{R}^{3} \mid v_{1} v_{2}=v_{3}\right\\}\) (d) \(U_{4}:=\left\\{\left(\begin{array}{l}v_{1} \\ v_{2} \\\ v_{3}\end{array}\right) \in \mathbb{R}^{3} \mid v_{1}=v_{2}\right.\) oder \(\left.v_{1}=v_{3}\right\\}\)

Gegeben sind ein Untervektorraum \(U\) eines \(\mathbb{K}-\) Vektorraums \(V\) und Elemente \(\boldsymbol{u}, \boldsymbol{w} \in V\). Welche der folgenden Aussagen sind richtig? (a) Sind \(u\) und \(w\) nicht in \(U\), so ist auch \(u+w\) nicht in \(U\). (b) Sind \(u\) und \(w\) nicht in \(U\), so ist \(u+w\) in \(U\). (c) Ist \(u\) in \(U\), nicht aber \(\boldsymbol{w}\), so ist \(\boldsymbol{u}+\boldsymbol{w}\) nicht in \(U\).

Zeigen Sie, dass die Menge \(\mathbb{K}^{m \times n}\). aller \(m \times n\) Matrizen 眉ber einem K枚rper \(\mathbb{K}\) mit komponentenweiser Addition und skalarer Multiplikation einen \(\mathbb{K}\)-Vektorraum bildet.
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