Electric Potential: Videos & Practice Problems

Two hollow plastic globes of radius 4.50 cm and 8.40 cm are concentric with each other. The globe of radius 4.50cm has a charge of -12.5 nC distributed evenly on its surface. Similarly, the other globe has a charge of +8.25 nC evenly distributed over its surface. If the electric potential is zero at an infinitely large distance, determine the resultant electric potential from the two globes at i) r = 0 cm ii) r = 6.00 cm iii) r = 8.40 cm.

An experimental setup consists of two charged particles K = +1.20 µC and L = -3.60 µC. Their separation is 29.0 cm. Assuming the electric potential is zero at infinity, determine the electric potential at a point that is 20.0 cm from K and 21.0 cm from L. 

During an experiment, charged particles M and N are placed 65.0 cm apart. Their charges are M = -8.50 µC and N = +3.50 µC. If the electric potential is zero at infinity, determine the electric potential at a point between the two charges, 30.0 cm from charge M. 

Two charged particles with a separation x have the same kind of charge and equal magnitude. Considering an axis joining the two particles, determine all points on the axis where the electric field is zero. Will the electric potential be zero at the points you identified?

Two charged particles carry the same kind of change and the same magnitude. Their separation is x. For an axis passing through both particles, determine all points where the electric potential is zero (Take the electric potential to be zero at infinity). Will the electric field be zero at the points you have identified? 

A rectangle measures 0.0400 m by 0.0500 m. Particles of charges +4.00 nC and -4.00 nC are placed on two adjacent corners separated by the shorter length. What is the electric potential at the center of the rectangle due to the two charges? Take potential to be zero at infinity.

Two particles charged to +3.80 μC and +5.50 μC are fixed in place 0.400 m apart. A proton is dropped at rest at the midpoint of the two charges. If the proton moves purely along the line joining the two charged particles, determine its speed when at a distance of 18.0 cm from the +3.80 μC particle.

A test charge carries a charge of -7.20 μC and has a mass of 0.240 g. It is moved from point P to point Q. The potential at P and Q is +125 V and 584 V respectively. If the net force on the test charge is the electric force and the charge has a speed of 2.40 m/s at point P, calculate its speed at point Q.

A very long horizontal wire has a uniform linear charge density of 92.0 µC/m. A radioactive nuclide emits an alpha particle (mass = 6.64 × 10^-27 kg, charge = +2e) toward the wire. Its speed is 1.4 × 10⁷ m/s when at a position 28.0 cm from the wire. What is the kinetic energy of the alpha particle? 

A charged particle lies at the origin of a xy plane. The electric field magnitude at point p from the origin is 27.4 V/m while the electric potential is 14.1 V. What is the direction of the electric field at point p? Take the electric potential to be zero at infinity. 

Students place a charged particle at point a. They find the electric field magnitude to be 84.6 V/m while the potential is 50.0 V at point b. Determine the magnitude of the charge on the particle if the potential is zero at infinity.

The electric potential at point p from a charged particle is 10.2 V. The electric field at that same point has a magnitude of 22.5 V/m. Calculate the length from the charge to point p. Assume potential is zero at infinity.

A proton has a speed 3.5 × 10⁶ m/s. The speed of the proton should be increased to 7.5 × 10⁶ m/s. Determine the potential difference along the proton's path that will achieve this increase. 

Two equal giant sheets of metal are placed parallel to each other with a separation of 54 mm. The sheets are charged by induction, charging them with opposites charges to a charge density of magnitude 20.2 nC/m². The separation of the sheets is tripled while the charge density is kept the same. Determine the change in the electric field and the potential difference between the plates. 

a) A steel ball has a radius of 20.0 cm. The electric potential of the ball's center relative to infinity is 1.2 kV. Determine the value of the unbalanced charge on the ball. b) Determine the potential at the surface of the ball relative to infinity. 

Consider two point charges: one carrying a negative charge of -2.0 nC and the other carrying a positive charge of +5.0 nC. Assume the positive charge is located at x = 0 cm, and the negative charge is at x = 2.0 cm on the x-axis. At what point or points on the x-axis does the electric potential become zero?

Consider a positively charged sphere in space with a radius of 0.5 mm. Two points are measured from the center of the sphere, one at 3.0 mm and another at 6.0 mm. A voltage difference of 400 V exists between these two points. Determine the sphere's charge.

Using the expression for electric potential at a point P located at a perpendicular distance R from a thin wire of length L with a uniform linear charge density λ as shown below, determine the electric field E at point P. $V_{P}=\frac{\lambda}{4\pi\epsilon_0}\ln\left(\frac{\frac{L}{2}+\sqrt{R^2+\left(\frac{L}{2}\right)^2}}{-\frac{L}{2}+\sqrt{R^2+\left(\frac{L}{2}\right)^2}}\right)$

The figure below shows an Electric field (V/m) vs position (m) graph. Determine the magnitude of potential difference (ΔV) between the positions x = 0m and x = 3m.

The provided diagram illustrates an arrangement of two uniformly charged spherical balls. Determine the potential difference (V_M-V_N) between positions M and N. Remember that at any given point, the potential is the combined effect of the potentials generated by all the charges present.

The contour map of electric potential is shown below, with a 2 cm × 2 cm grid overlaid on it for reference. Find the strength and orientation of the electric field in the x-direction at points A and B. Assume the field is constant between the equipotential surfaces.

The graph below represents the electric potential versus distance along the y-axis. The electric potential is uniform along the x and z directions. Find the electric field strength E_y at (i) y = -4 mm, (ii) y = 1 mm, and (iii) y = 4 mm.

Given the equations, complete the solution 

(9.0 × 10⁹ Nm²/C²) [((Q₁)(4Q₂)) /(5.0 × 10^-3 m)] = 2.0 × 10^-6 J

Q₂ = 4Q₁

Two point charges are located on the x-axis. The first charge is -q and is located at -d, while the second charge is +q and is located at +d. Derive an expression for the components of the electric field vector E along the x-axis and y-axis due to the two charges where d << x and y.

At a scientific institute, researchers have recently uncovered an empirical equation that describes the electric potential between two electrodes: V(x) = 100 ln(1 + x/5). Here, x is measured in meters, and it is the distance originating from the first electrode and extending towards the second electrode. The separation between the electrodes is 10 meters. Determine the magnitude of the electric field strength when positioned 3.5 meters away from the first electrode.

What is the magnitude and direction (measured as an angle from the positive x-axis), of the electric field at the coordinates (3.0 m, -1.0 m) in a specific region of space with an electric potential described by V = (175x² − 280y²) V, where x and y are measured in meters?

Three-point charges are located on the x-axis at x = -a, 0, and a. The charge at the origin is negative and has a magnitude of q, while the other two charges are positive and have a magnitude of 2q each. Find an expression for the electric field at a point P located on the y-axis, such that y << a.

Consider an electric dipole made up of two point charges, each with magnitude q, but of opposite signs. These charges are separated by a distance of 2d along the x-axis. Specifically, the positive charge is situated at x=d, and the negative charge is at x=-d. The figure below presents a graph of the electric potential V as a function of x along this axis. Sketch a corresponding graph of E_x, which represents the x-component of the electric field.

Consider two point charges, one carrying a charge of -20.0 nC and the other carrying a charge of +40.0 nC. These charges are located 14.0 cm apart on the x-axis. Calculate the electric potential at the point on the x-axis where the electric field is zero.

In the figure below, which point P or Q possesses a greater electric potential?

In a quantum mechanical model of the hydrogen atom, an electron is found to be in an excited state. At this state, the electron is located at a distance of 0.127 nm from the nucleus. Determine the electric potential experienced by the electron.

An electron is in an excited state in a Bohr model of the hydrogen atom. At this state, the electron is located at a distance of 0.076 nm from the nucleus. Determine the potential energy of the electron at this distance.

Determine the value of the electric potential at the point marked in green in the figure below.

The graph represents the electric potential for the two-point charges situated at positions x = m and x = n, respectively, along the x-axis. Determine the ratio of their magnitudes, |q_m/q_n|.

An isolated metallic sphere carries an electric charge. Measurements taken from the sphere's surface indicate that the electric potential is 9000 Volts at a distance of 1.4 cm and 3000 Volts at a distance of 4.5 cm. Determine the radius of the sphere.

During a nanotech experiment, a scientist has two tiny mercury spheres, each bearing a 0.20 nC charge and having a surface potential of 100 V. Upon merging these spheres into a single larger one, what would the potential at the surface of the combined sphere be?

A metallic rod of length L and radius R is charged to a potential of 800 V. A second metallic rod of length 2L and radius 2R is charged to a potential of 1600 V. The rods are placed end-to-end, such that their flat circular faces are in contact, and then separated. Calculate the potential of each rod after they have been separated. Note: R << L.

A charged plastic rod of length L has a linear charge density given by λ(x) = λ ₀(1-x/L) along its length, where x is measured from one end of the rod. Determine the electric potential at a point P located on the axis at a distance 10L from the center of the rod.

Consider that the initial y-coordinate is 8.0 cm, the final y-coordinate is 45 cm, and the uniform electric field E_y has a strength of 4000 V/m. What would be the potential difference between these two points?

The figure below shows an Electric field (V/m) vs position (m) graph. Determine the magnitude of potential difference (ΔV) between the positions x = 0m and x = 3m.

What is the magnitude of the potential difference, in volts, between two positions, z_i = -10 cm and z_f = 20 cm, when subjected to an electric field described by E_z = e^2z V/m?

Determine the electric field and the electric potential at the coordinates (x, y) = (4.0 m, 2.0 m). You are given a region of space where the electric field is characterized by the following equation: Ē = (200x î + 400y ĵ) V/m, where x and y are expressed in meters. It is important to note that the electric potential is set to zero at the origin and the potential difference remains the same regardless of the path taken to connect two points.

A charged cylindrical insulator of radius R and length L carries a uniform charge density λ as illustrated below. Determine the electric potential at a distance r from the center of the cylinder along its axis. Note that R << L. 

Consider a thin rod with an insulating property and a charge Q. This rod is shaped into a quarter-circle with a radius of R, as illustrated below. Formulate an equation for the electric potential at the center of this quarter-circle. 

Consider a region with a uniform electric field (E = 2250 V/m) as shown in the figure. Determine the potential difference between the two points P and Q horizontally 12cm apart.

A uniform charge density ρ is present throughout the volume of a lengthy cylindrical object with diameter D. Determine the potential difference between the core axis of the cylinder and its outer perimeter.

Consider two points in space called P and Q. The electric potential at point P is 2.82 kV, and the electric potential at point Q is 1.25 kV. Calculate what the potential difference at: ΔV_QP = V_P - V_Q will be in units of kV.

Use the figure below to calculate the potential difference ΔV_CA.

The electric potential V_y as a function of distance y is shown below. Illustrate how the electric field E varies with distance y.

A very tall rod lying vertically has a linear charge density of 84.0 µC/m. An alpha particle (mass = 6.64 × 10^-27 kg, charge = +2e) emitted from a decaying nucleus moves toward the rod. Its speed is 1.8 × 10⁷ m/s  when it is 25.0 cm from the rod. Calculate the point where the alpha particle reaches before it is repelled back. 

An electric fence is constructed from an insulating rod of radius 2 cm. The rod has a homogenous linear charge density of 10 nC/m. Determine the distance of a point p measured from the surface of the rod, such that the potential between the point and the rod's surface is 150 V.

Two large sheets of metal lie parallel to each other with a separation of 3.20 cm. They are charged with opposite charges to the same surface charge density whose magnitude is 26.2 µC/m². Calculate the potential difference between the two sheets. 

To produce an electric field whose strength has the same order of magnitude as the Earth's magnetic field (such that E = 50 µV/m) in a parallel plate capacitor using a 5.0 V adapter, what plate separation would be required?

Determine the expression for the electric potential V at position x in a certain region of space, where the electric field is given by E_x = 3500x V/m. Consider a reference point at x = 0 where the electric potential is 1.5 V.

Find the expression for the electric potential V(x_A, y_A) at point A in the xy-plane, where an electric dipole is positioned at the origin with two charges +n•e and -n•e separated by a distance of d along the x-axis. Your response should be in terms of n•e, d, x_A, and y_A.

A small metal sphere with a radius of 0.8 cm has an excess of 2.5 × 10⁸ electrons. Determine the potential of the metal sphere.

Provide an expression for the electric potential on the y-axis at position y, resulting from two positive point charges q₁ and q₂ positioned on the x-axis at x = ±a.

Suppose you have a steel rod of length 6l. You place that rod horizontally in such a way that the rod is centered on the horizontal axis (x-axis). The steel rod uniformly carries a total positive charge of Q. What will be the potential V with respect to the vertical axis, considering as a function of y? Suppose that the potential approaches zero at infinity.

A thin, straight metal rod is bent to form a shape as shown below. The metal rod is uniformly charged with a linear charge density of λ. Determine the electric potential at point O.

Consider a hollow cylindrical shell constructed from PVC (Polyvinyl Chloride) plastic. The shell has a length L and a diameter D. This shell carries a total charge Q, distributed uniformly along the entire cylinder length. Determine the electric potential at the central point within this cylindrical shell.

Consider a scenario in which a uniformly charged sphere of radius R has a total charge of Q. The potential difference ΔV between the potential at the sphere's center, V₀, and a point at radius r from the center is: ΔV = -Q / (4πε_oR³) • (r² /2). Determine what the ratio V_o / V_R will be, where V_R represents the potential at the sphere's surface.

The solar panels of the Johnson family generated 8 MJ of energy and transferred a charge of 5 C in one day. What was the potential difference (voltage) across the solar panels?

A voltage signal can range from 0 V to 10.00 V, which is to be converted into a 10-bit binary representation. What binary number would best represent a voltage of 7.35 volts?

Calculate the amount of work that is generated by an electric field when it moves a charge of -7.1 μC from one location to another where an increase in a potential difference of +32 V occurs.

A physicist releases a proton into an electric field. Afterward, it gains 3.2 × 10^-16 J of kinetic energy as it moves from point X to point Y within the field. Determine the voltage difference between these two points and conclude which of these two points has a greater electric potential value.

In a special lab, measurements are conducted on a grain of sand floating in an electric field. It has a mass of 0.075 g and a charge of 3.0 × 10^-6 C, and the potential at its position x is V(x) = (3.0 V/m) x - (4.0 V/m²) x². If the grain starts at x = 3m, Determine its initial acceleration.

A physicist is trying to calculate the charge density on the sheets of a parallel plate capacitor. It is given that the electric potential of the field between the sheets can be represented by V(y) = (9 V/m)y + 6 V. Where y = 0 at the position of one of the sheets. Considering the positive direction of y is toward the other sheet, determine the magnitude of the charge density on these sheets.

A digital-to-analog converter operates with a range from 0V to 10V and uses an 8-bit system for conversion. What voltage corresponds to a binary representation of '10011011'?

A 3-bit analog-to-digital converter (ADC) is used in a system where the highest voltage is 4.0 V. What voltage does the binary number 110 represent?

Compute the electric potential caused by an infinitesimal dipole with a moment of magnitude equaling 6 × 10^-29 C·m at an observation location that is distanced by exactly 3 nm away from it and lies on its axial line but closer to its positive charge.[ ϵ₀​: 8.85 × 10⁻¹² C²/N⋅m²]

Find the electric potential at a point situated 3.2 × 10^-9 m away along the axis of an infinitesimal dipole having a dipole moment of 3.5 × 10^-30 C·m.

A total charge Q is evenly distributed over the surface of a thin flat disk of copper with radius R₀. The electric potential at a distance y along the y-axis perpendicular to the disk through the center is given by $V(y)=\frac{Q}{2 \pi \epsilon_{0} R_{0}^{2}}\left[\left(y^{2}+R_{0}^{2}\right)^{1 / 2}-y\right]$

What is the expression for the electric field E(y) at a distance y on the y-axis from the center of the disk?

Consider that an external force moves a stationary charge of -8.0 μC from point P to point Q. The work done during this process is 18 × 10^-4 J. Determine the voltage difference between points P and Q, if the charge gained a kinetic energy of about 7.0 × 10^-4 J upon reaching point Q.

Due to electrostatic effects, a long underground pipeline with a radius R₀ (where R₀ is much smaller than the length, $l$) carries a uniform surface charge density $\sigma$ (C/m²). The pipeline is maintained at a potential V₀. Determine the electric potential at a distance R from the center of the pipeline, which is far from its ends.

Analyze the following cases:

(i) R is greater than R₀

(ii) R is less than R₀

(iii) Is V equal to 0 at R equals infinity (assume the length is infinite)? Explain. [Hint: Recall Gauss’s law.]

A large metallic sphere used in an electrostatic machine must maintain a voltage of about 80,000 volts without discharging into its environment. What must be its minimum radius? Also, calculate how much charge it carries.

A conducting ball with a diameter of 25 cm has been charged such that it has an electrical potential difference relative to infinity equal to +600 V. (i) What would be its surface charge density? (ii) At what distance from the center will the electric potential due to the sphere be +50 V?

You have a wire of length L with a uniform charge Q distributed along it. You bend this wire into a quarter circle and calculate the electric potential at the center of the full circle. Assume the electric potential V is zero at infinity.

Suppose you have two-point charges q₁ and q₂ placed 10 cm apart on the y-axis. One charge is q₁ = 4.5 μC (positive), and the other is q₂ = 3.0 μC (negative). Your task is to find the points along the y-axis where the electric field is zero.

Find the breakdown voltage in the air for a small charged metallic sphere of radius 1.0 cm. Additionally, determine the surface charge density on the sphere at this voltage.

Calculate the electric potential due to a nucleus at the location of an electron orbiting the nucleus with a single proton in a circular path. The radius of the orbit is 0.26 × 10^-9 m.

In the Unicode system, characters are represented as hexadecimal numbers. If the letter 'F' is represented by '0046', what would be the representations for 'H' and 'P'?

In the Unicode system, characters are represented as hexadecimal numbers. The code for the character 'a' is 0061, and the next characters are just the next hexadecimal numbers. What would be the decimal numbers for Unicode 'b' and 'k'?

In the diagram, a uniform electric field $\overrightarrow{E}$ = (8.50 N/C)𝒊̂ is directed along the positive x-axis. Given the coordinates of points P, Q, R, and S (in meters), calculate the potential difference between points P & Q.

As shown in the diagram, a uniform electric field $\overrightarrow{E}$ = (5.20 N/C)𝒋̂ is directed along the positive y-axis. Given the x and y coordinates of points D, E and F (in meters), determine the potential difference V_DF.

A point charge of +4.5μC is located at the origin. Point P is positioned 40.0 cm to the east of this charge, while point R is located 30.0 cm to the south. Calculate the difference in electric potential between points P and R, denoted as V_PR.

Given that V = P / (x² + U²)³ where P is equal to 300 V•m⁶ and U equals 0.35 m. Evaluate (i) V when x = 0.35 m. Also, (ii) calculate E̅ as a function of x. Then finally (iii) determine E̅ when x equals 0.35 m.

Inside a high-voltage lab, engineers have designed a storage container for electrical energy using a nonconducting sphere of radius r_B that contains a concentric spherical cavity of radius r_A. The uniform charge density ρ_E (C/m³) is held by the dielectric material that fills the space between r_A and r_B. With the reference set at V = 0 when r = ∞, derive an expression for the electric potential V as a function of the distance r from the center for r > r_B.

Consider a lab experiment in which visible light dislodges electrons from a caesium metal surface. A collector plate, charged negatively and placed above the caesium, is set to a potential that halts the fastest electrons, forcing them to return to the surface. If the collector plate voltage is −1.85V relative to the caesium when these electrons are stopped, find their speed at the emission time.

A charged spherical conductor can reach very high electric potentials before the electrical breakdown of the surrounding air occurs. Consider a spherical conductor with a radius of 0.25 m that can withstand an electric field of up to 4.0 × 10⁶ V/m before breakdown occurs. What is the maximum electric potential on the surface of the sphere just before electrical breakdown? Assume the potential V = 0 at r = ∞.

Suppose you have two-point charges placed 10 cm apart on the y-axis. One charge is 4.5 μC (positive), and the other is 3.0 μC (negative). Your task is to find the points along the y-axis where the electric potential is zero, and for this, assuming that the potential is zero at infinity.

At points along the z-axis, determine the electric potential V(z), relative to V = 0 at $z=\infty$, due to a flat circular disk with a nonuniform surface charge density σ = k r². Here, r is the radial distance from the center of the disk, k is a constant, and the disk has a radius R.

In a hydrogen sulfide (H₂S) molecule, the two H-S bonds form an angle of 92°. The measured dipole moment is p = 0.97 x 10⁻³⁰ C·m, and the bond length between hydrogen and sulfur is d = 1.336 × 10⁻¹⁰ m. Determine the effective charge q on each hydrogen atom.

For a hydrogen sulfide (H₂S) molecule, the two H-S bonds form an angle of 92°, and the bond length between each hydrogen and sulfur atom is 1.34 × 10⁻¹⁰ m. The molecule has a dipole moment of 0.97 × 10⁻³⁰ C·m. Determine the expression for the electric potential 𝑉, at point P which is far from the molecule, caused by the dipole moments 𝑝₁ and 𝑝₂ of the two H-S bonds. Hint:You can use the following relation to calculate 𝑝₁​ and 𝑝₂:

$p_{1}=p_{2}=\frac{p}{2 \cos \left(92^{\circ} / 2\right)}$ where $p$ is the net dipole moment of the molecule.

A Van de Graaff generator is used in a research laboratory to study high-energy particle interactions. The generator consists of a large spherical metal shell with a radius of 0.30 m as shown. Over time, as the generator operates, the spherical shell becomes highly charged, resulting in a significant electric potential at its surface. researchers measure the potential at the surface of the sphere to be 8.0 × 10⁵ volts. Determine the amount of charge accumulated on the surface of the spherical shell. Hint: Coulomb's constant, k = 8.99×10⁹ N⋅m²/C².