Area = πr²; 50π = πr²; r² = 50; r = √50; Diameter = 2√50 ≈ 10 units.

The dielectric constant (κ) of a vacuum is defined as 1, which serves as the reference for other materials.

The dimension of energy is M^1L^2T^-2.

The dimension of force is M^1L^1T^-2.

Conventional current is defined as the flow of positive charge, which is from positive to negative.

The direction of the magnetic field around a straight conductor carrying current is counterclockwise when viewed from the positive end of the conductor.

Using the right-hand rule, if the current flows in a counterclockwise direction, the magnetic field inside the loop points out of the plane.

The magnetic field inside a current-carrying solenoid is directed from the north pole to the south pole of the solenoid.

The discriminant is b² - 4ac = (-12)² - 4*3*12 = 144 - 144 = 0.

The discriminant is b² - 4ac = (-12)² - 4*3*9 = 0.

The distance from the center to a point on the circumference is the radius, which is 12 cm.

The distance from the center to a point on the circumference is the radius, which is 8 cm.

Distance = |d1 - d2| / √(A² + B² + C²) = |5 - 10| / √(2² + 3² + (-1)²) = 5/√14.

Distance = √[(2 - (-1))² + (3 - (-1))²] = √[3² + 4²] = √25 = 5.

Distance = √[(2 - (-1))² + (3 - (-1))²] = √[3² + 4²] = √25 = 5.

Using the distance formula, d = √((5 - 2)² + (7 - 3)²) = √(9 + 16) = √25 = 5.

Distance = √[(7-3)² + (1-4)²] = √[4 + 9] = √13 ≈ 5.

Distance = √[(4-1)² + (5-2)² + (6-3)²] = √[3² + 3² + 3²] = √27 = 3√3.

Distance = |1 + 1 - 2| / √(1² + 1²) = 0 / √2 = 0.

Distance = |1 + 2 - 3| / √(1² + 1²) = |0| / √2 = 0, closest option is 1.

Distance = |1 + 2 - 5| / √(1² + 1²) = | -2 | / √2 = 2/√2 = √2.

Distance = |3 + 4 - 5| / √(1² + 1²) = |2| / √2 = 2/√2 = √2.

A · B = (2)(4) + (3)(-1) = 8 - 3 = 5.

Inserting a dielectric material increases the capacitance of a capacitor by reducing the electric field within it.

Adding a common ion decreases the solubility of a salt due to the common ion effect, which shifts the equilibrium to the left.

Adding a non-volatile solute increases the boiling point due to boiling point elevation.

Adding a non-volatile solute decreases the vapor pressure of the solvent due to the solute particles occupying space at the surface.

Cortisol is a steroid hormone that helps reduce inflammation and is involved in the stress response.

Deforestation leads to habitat loss, which significantly decreases biodiversity as many species lose their homes.

Deforestation increases carbon dioxide levels in the atmosphere as trees that absorb CO2 are removed.