Identify the relationship between Y (electric potential) and X (distance from the surface of an isolated charged sphere) and select the correct graph.

Here are the solutions to the questions you've provided: Question 48: Y: Electric potential X: Distance from the surface of an isolated charged sphere Step 1: Recall the formula for electric potential. For an isolated charged sphere with charge Q and radius R, the electric potential V at a distance r from its center (for r R) is given by: V = (kQ)/(r) where k is Coulomb's constant. Step 2: Relate distance from the surface to distance from the center. Let X be the distance from the surface of the sphere. The distance from the center of the sphere is r = R + X. Step 3: Substitute into the potential formula. Y = V = (kQ)/(R+X) Step 4: Analyze the relationship. This equation shows that the electric potential Y is inversely proportional to (R+X). As X increases, Y decreases, approaching zero asymptotically. This is a characteristic inverse curve. Step 5: Match with the graphs. Graph A shows a curve that starts at a high positive value (when X=0, Y=kQ/R) and decreases as X increases, approaching zero. This matches the derived relationship. Answer for 48: A Question 49: Y: The radius of the orbit of an electron moving at constant speed in a perpendicular uniform magnetic field X: The momentum of the electron Step 1: Identify the forces acting on the electron. When an electron with charge q and mass m moves with speed v in a uniform magnetic field B perpendicular to its velocity, the magnetic force F_B provides the centripetal force F_c. Step 2: Write the formulas for magnetic and centripetal forces. Magnetic force: F_B = qvB Centripetal force: F_c = (mv^2)/(r) where r is the radius of the orbit. Step 3: Equate the forces and solve for r. qvB = (mv^2)/(r) r = (mv)/(qB) Step 4: Relate to momentum. Momentum p is defined as p = mv. So, Y = r = (p)/(qB). Step 5: Analyze the relationship. Since q and B are constants (charge of electron and uniform magnetic field), the radius Y is directly proportional to the momentum X. This is a linear relationship passing through the origin (Y=0 when X=0). Step 6: Match with the graphs. Graph B shows a straight line passing through the origin with a positive slope, indicating a direct proportionality. Answer for 49: B Question 50: Y: Reactance of an inductor X: Frequency Step 1: Recall the formula for inductive reactance. The inductive reactance X_L of an inductor with inductance L at a frequency f is given by: X_L = 2 f L Step 2: Relate to Y and X. Here, Y = X_L and X = f. So, Y = 2 L X. Step 3: Analyze the relationship. Since 2 L is a constant (for a given inductor), the inductive reactance Y is directly proportional to the frequency X. This is a linear relationship passing through the origin (Y=0 when X=0). Step 4: Match with the graphs. Graph B shows a straight line passing through the origin with a positive slope, indicating a direct proportionality. Answer for 50: B That's 2 down. 3 left today — send the next one.