Ch 25: The Electric Potential

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 25: The Electric Potential

Problem 51

Chapter 25, Problem 51

What is the escape speed of an electron launched from the surface of a 1.0-cm-diameter glass sphere that has been charged to 10 nC?

Verified step by step guidance

Determine the formula for escape speed. The escape speed \( v_{\text{escape}} \) is given by \( v_{\text{escape}} = \sqrt{\frac{2k_e Q}{m r}} \), where \( k_e \) is Coulomb's constant (\( 8.99 \times 10^9 \ \text{N·m}^2/\text{C}^2 \)), \( Q \) is the charge of the sphere, \( m \) is the mass of the electron (\( 9.11 \times 10^{-31} \ \text{kg} \)), and \( r \) is the radius of the sphere.

Convert the diameter of the sphere to its radius. The diameter is given as 1.0 cm, so the radius \( r \) is \( \frac{1.0}{2} \ \text{cm} = 0.5 \ \text{cm} = 0.005 \ \text{m} \).

Substitute the given values into the formula. Use \( Q = 10 \ \text{nC} = 10 \times 10^{-9} \ \text{C} \), \( r = 0.005 \ \text{m} \), \( m = 9.11 \times 10^{-31} \ \text{kg} \), and \( k_e = 8.99 \times 10^9 \ \text{N·m}^2/\text{C}^2 \).

Simplify the expression under the square root. Calculate \( \frac{2k_e Q}{m r} \) by multiplying and dividing the constants and variables appropriately.

Take the square root of the result from the previous step to find the escape speed \( v_{\text{escape}} \). Ensure the units are consistent and the final speed is in meters per second (m/s).

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Key Concepts

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

Escape Velocity

Escape velocity is the minimum speed an object must reach to break free from the gravitational influence of a celestial body without further propulsion. It depends on the mass of the body and the distance from its center. For small objects, like an electron, the escape velocity can be calculated using the formula v = √(2GM/r), where G is the gravitational constant, M is the mass of the body, and r is the radius.

Electric Field and Potential

When a charged object, such as the glass sphere in this scenario, creates an electric field around it, this field exerts a force on other charged particles, like electrons. The electric potential energy associated with this field can influence the escape speed of the electron. The potential energy can be calculated using the formula U = kQq/r, where k is Coulomb's constant, Q is the charge of the sphere, q is the charge of the electron, and r is the distance from the center of the sphere.

Coulomb's Law

Coulomb's Law describes the force between two charged objects. It states that the force (F) between two charges is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them. This law is crucial for understanding how the charged glass sphere affects the electron's motion and helps determine the energy required for the electron to escape the sphere's electric field.

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Coulomb's Law

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A room with 3.0-m-high ceilings has a metal plate on the floor with V=0 V and a separate metal plate on the ceiling. A 1.0 g glass ball charged to +4.9 nC is shot straight up at 5.0 m/s. How high does the ball go if the ceiling voltage is +3.0×10⁶ V?

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