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Wize University Physics Textbook (Master) > Newton's Laws of Motion: Forces and Dynamics

Electrostatic Force

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Electrostatic Force


  • Electric forces between charged objects:
DIRECTION OF FORCE: Opposite charges attract each other and same charges repel each other!


MAGNITUDE OF FORCE: from Coulomb's Law

Coulomb’s law

  • The force between to charged particles is proportional to the product of the two charges, and inversely proportional to the distance squared.
Fe=ke∣q1∣∣q2∣r2F_e=k_e\frac{|q_1||q_2|}{r^2}
where q1 and q2 correspond to the two charges, and ke is a proportionality constant equal to 8.99 x 109 N*m2*C-2.

  • Note that this equation only gives the magnitude of the force, hence the absolute values on the charges (so that the result is positive).
  • The direction of the force is determined by drawing a line joining both charges, and then deciding whether the force is attractive or repulsive. Then, the force is drawn accordingly along that line.

  • The superposition principle states that if there are more than two charged objects, the total force on one of these is obtained by calculating each individual force on that object (by taking the charges two by two) and then adding all the forces together.








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Electric Field

A charged particle with charge q, creates an electric field in their surrounding space.


  • The electric field is a vector field: for each point in space, you can define a vector quantity corresponding to that field.
  • It will have a magnitude and a direction.

  • The magnitude of the electric field created by a single point charge q is given by this formula:
E=ke∣q∣r2E=k_e\frac{|q|}{r^2}

  • As for the direction of the electric field, it is radial, but we use the following convention: it comes out of a positive charge, and goes into a negative charge






  • If we introduce a test charged particle with charge qo in the region with an electric field, an electrostatic force (Fe) will be exerted on the test particle due to the presence of the electric field.
Fe=ke∣q∣∣qo∣r2Fe→=qE⃗\begin{array}{c} \displaystyle F_e=k_e\frac{|q||q_o|}{r^2}\\[11pt] \stackrel{\rightarrow}{F_e}=q\vec E \end{array}

  • Electric force is radial, meaning that it acts along the line joining two charges.

  • The force on a positive test charge particle is in the same direction of the electric field and the force on a negative charge particle is opposite the electric field direction.

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Two point charges are located as shown below. Where should be a third charge q3=8μCq_3=8\mu C placed on the x-axis so that the net force on q1q_1 isF1=8NF_1=8N in the negative x- direction?
Solution:
Since FnetF_{net}on q1q_1, is toward negative x and q3q_3 is a positive charge , q3q_3 should be on the right of q1q_1!

→ ∣F⃗13∣−∣F⃗12∣=∣Fnet∣\rightarrow\ \left|\vec{F}_{13}\right|-\left|\vec{F}_{12}\right|=\left|F_{net}\right|
=∣k q1q3r13 2∣−∣k q1q2r12 2∣=8=\left|\frac{k\ q_1q_3}{r_{13}^{\ 2}}\right|-\left|\frac{k\ q_1q_2}{r_{12}^{\ 2}}\right|=8

→ r132=∣k q1 q3∣8+∣k q1q2r122∣=9×109×2×8×10−128+9×109×2×6×10−12(0.3 )2=0.0157\rightarrow\ r_{13}^2=\frac{\left|k\ q_1\ q_3\right|}{8+\left|\frac{k\ q_1q_2}{r_{12}^2}\right|}=\frac{9\times10^9\times2\times8\times10^{^{-12}}}{8+\frac{9\times10^9\times2\times6\times10^{-12}}{\left(0.3\ \right)^2}}=0.0157

→ r13=0.0157=0.125 m\rightarrow\ r_{13}=\sqrt{0.0157}=0.125\ mon the right of q1q_1
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A charged ball is placed 1m away from a sphere with a +1nC charge. How far away from a sphere with a +4nC charge would it have to be placed to be subjected to the same force?
Extra Practice
Conductors, Insulators and Induced Charges
Practice: Conductors, Insulators, and Induced Charge
Consider two spheres; they are either positively charged, negatively charged, or neutral.
a) What can you say about the charge on the two spheres if they are electrically attracted to each other?
Coulomb’s Law
Two point charges are located on the x-axis as shown below.
a) Where should a third charge, q3=+8.00μCq_3=+8.00\mu C, be placed on the y-axis so that the net force on q2q_2 makes an angle of +45°+45\degree above the negative x-axis?
b) For additional practice: What is the magnitude of the net force? (Not in video; see written solution)
Square of charges
Four charges are arranged in a square with sides of length 2.5 cm.
Find the net force on q1 (both magnitude and direction)
Kinematic equations with a charge
An electron is shot at an initial speed of 3.00 × 106 m/s, perpendicular to a uniform electric field, as shown below. Electrons hits the right plate at vertical distance 6.00 cm.
a) If the electron was 4.00 cm away from this plate initially, calculate the magnitude of the electric field.
b) What is the direction?
Coulomb’s Law
EX: Two point charges are located on the x-axis as shown below.
a) Where should a third charge, q3=+8.00μCq_3=+8.00\mu C, be placed on the y-axis so that the net force on q2q_2 makes an angle of +45°+45\degree above the negative x-axis?
b) For additional practice: What is the magnitude of the net force? (Not in video; see written solution)
Placing a third charge - x-axis
EX: Two point charges are located as shown below.
a) Where should be a third negative chargeq3q_3be placed on the x-axis such that the net forces onq1q_1 andq2q_2 be equal to zero?
b) What should the magnitude of the charge be?
Conductors, Insulators and Induced Charges
Practice: Conductors, Insulators, and Induced Charge
Consider two spheres; they are either positively charged, negatively charged, or neutral.
a) What can you say about the charge on the two spheres if they are electrically attracted to each other?
Placing a third charge - x-axis
Two point charges are located as shown below.
a) Where should be a third negative charge q3q_3 be placed on the x-axis such that the net forces on q1q_1 and q2q_2 be equal to zero?
b) What should the magnitude and sign of the charge be?
Coulomb’s Law
Two point charges are located on the x-axis as shown below.
a) Where should a third charge, q3=+8.00μCq_3=+8.00\mu C, be placed on the y-axis so that the net force on q2q_2 makes an angle of +45°+45\degree above the negative x-axis?
b) For additional practice: What is the magnitude of the net force? (Not in video; see written solution)
Square of charges
Four charges are arranged in a square with sides of length 2.5 cm.
Find the net force on q1 (both magnitude and direction)
Placing a third charge - y-axis
Practice: Coulomb's Law with Vectors
Two point charges are located on the x-axis as shown below.
a) Where should a third charge, q3=+8.00μCq_3=+8.00\mu C, be placed on the y-axis so that the net force on q2q_2 makes an angle of +45°+45\degree above the negative x-axis?
Conductors, Insulators and Induced Charges
Practice: Conductors, Insulators, and Induced Charge
Consider two spheres; they are either positively charged, negatively charged, or neutral.
a) What can you say about the charge on the two spheres if they are electrically attracted to each other?
Kinematic equations with a charged particle
Practice: Electron Moving Between Two Plates
An electron is shot at an initial speed of 3.00 × 106 m/s up the page. There is a uniform electric field that points from one parallel plate to the other, and the electron starts moving towards the plate on the right, as shown below. The electron hits the right plate at vertical distance of 6.00 cm above where it started.
a) Based on the direction of the electron motion shown below, what is the direction of the electric field?
Kinematic equations with a charged particle
Practice: Electron Moving Between Two Plates
An electron is shot at an initial speed of 3.00 × 106 m/s up the page. There is a uniform electric field that points from one parallel plate to the other, and the electron starts moving towards the plate on the right, as shown below. The electron hits the right plate at vertical distance of 6.00 cm above where it started.
a) Based on the direction of the electron motion shown below, what is the direction of the electric field?
Placing a third charge - y-axis
Practice: Coulomb's Law with Vectors
Two point charges are located on the x-axis as shown below.
a) Where should a third charge, q3=+8.00μCq_3=+8.00\mu C, be placed on the y-axis so that the net force on q2q_2 makes an angle of +45°+45\degree above the negative x-axis?
Square of charges
Four charges are arranged in a square with sides of length 2.5 cm.
Find the net force on q1 (both magnitude and direction).
...........
Conductors, Insulators and Induced Charges
Example: Conductors, Insulators, and Induced Charge
Consider two spheres; they are either positively charged, negatively charged, or neutral.
a) What can you say about the charge on the two spheres if they are electrically attracted to each other?
An electron (me = 9.109 x 10-31 kg) travels due north at a speed of 2.0 x 105 m/s. A constant electric field stops the electron in a distance of 10m. What is its magnitude and orientation?
At a certain point on Earth, there is a constant electric field E pointing downwards. A small ball of negligible mass carrying a charge of q is thrown straight upwards with a velocity v0. How high does the ball get if: a) q is positive? b) q is negative?
An ion with charge of 3.2 x 10-19 C and mass of 6.64 x 10-27 kg starts in a region with E=500 N/C k^\hat k . It moves for Δz=3m k^\Delta z=3m\,\hat k and then experiences a constant magnetic field.
With what speed does it enter the magnetic field region?
What magnetic field is required so that the ion continues with a constant velocity?
Placing a third charge - x-axis
Two point charges are located as shown below.
a) Where should be a third negative chargeq3q_3be placed on the x-axis such that the net forces onq1q_1 andq2q_2 be equal to zero?
b) What should the magnitude of the charge be?
Square of charges
Four charges are arranged in a square with sides of length 2.5 cm.
Find the net force on q1 (both magnitude and direction)
Coulomb’s Law
Two point charges are located on the x-axis as shown below.
a) Where should a third charge, q3=+8.00μCq_3=+8.00\mu C, be placed on the y-axis so that the net force on q2q_2 makes an angle of +45°+45\degree above the negative x-axis?
b) For additional practice: What is the magnitude of the net force? (Not in video; see written solution)
Conductors, Insulators and Induced Charges
Consider two spheres; they are either positively charged, negatively charged, or neutral.
a) What can you say about the charge on the two spheres if they are electrically attracted to each other?
(NB: In the video, I give one possible case - what are the other possible cases? See written solution!)
Kinematic equations with a charge
An electron is shot at an initial speed of 3.00 × 106 m/s, perpendicular to a uniform electric field, as shown below. Electrons hits the right plate at vertical distance 6.00 cm.
a) If the electron was 4.00 cm away from this plate initially, calculate the magnitude of the electric field.
b) What is the direction?