1) To determine the direction of the magnetic field for which the magnetic force on the wire is maximized, you can apply the right-hand rule. The right-hand rule states that if you point your thumb in the direction of the current flow (positive y-direction in this case), and then curl your fingers in the direction of the magnetic field, your palm will face the direction of the magnetic force.
Since the magnetic force (FB) is in the negative x-direction, your palm should be facing in the direction opposite to the magnetic force. In this case, it means that the magnetic field (B) should be in the positive x-direction. Therefore, the correct answer is a) positive x-direction.
2) To calculate the ratio of the magnetic field inside solenoid 1 to solenoid 2, you can use the formula for the magnetic field inside a solenoid, given by B = μ₀ * N * I / L, where μ₀ is the permeability of free space, N is the number of turns, I is the current, and L is the length.
Let's denote the subscripts "1" and "2" for solenoid 1 and solenoid 2 respectively.
Given the following ratios:
- N₁ / N₂ = 15
- r₁ / r₂ = 1/9
- I₁ / I₂ = 7 (I₁ = 7 * I₂)
The formula for the magnetic field can be rewritten as:
B₁ / B₂ = (μ₀ * N₁ * I₁) / (μ₀ * N₂ * I₂)
Since μ₀, a constant, is the same for both solenoids, we can cancel it from the equation:
B₁ / B₂ = (N₁ * I₁) / (N₂ * I₂)
Now substitute the given ratios:
B₁ / B₂ = (15 * 7) / (1 * 1)
B₁ / B₂ = 105
Therefore, the ratio of the magnetic field inside solenoid 1 to solenoid 2 is 105. Hence, the correct answer is a) 105.