A long wire carries a current of 20 A along the directed axis of a long solenoid. The field due to the solenoid is 4 mT. Find the field at a point 3 mm from the solenoid axis.
Magnetic field due to the long wire
B₁=μ₀I/2πb=4π•10⁻⁷•20/2•π•0.003=1.3•10⁻³ T
( B₁ is directed normal to the axis of solenoid)
B=sqrt(B₁²+B₂²) =sqrt{(1.3•10⁻³)²+(4•10⁻³)²} =4.2•10⁻³ T
Well, well, well, looks like we have a shocking question here! Let me use my electrifying humor to help you out.
So, you've got a long wire carrying a current? That's quite shocking, I must say! Now, this wire happens to be inside a solenoid, which can create a magnetic field, let's call it B. You want to know the field at a point 3 mm from the solenoid axis? Alrighty then!
First things first, we need to use the formula to calculate the magnetic field inside a solenoid, which is given by B = μ₀ * n * I. Where B is the magnetic field, μ₀ is the permeability of free space, n is the number of turns per unit length, and I is the current flowing through the solenoid.
Now, the problem states that the field due to the solenoid is 4 mT. So let's plug in the values. We know B = 4 mT = 4 × 10⁻³ T and I = 20 A.
Next, we need to find the number of turns per unit length, which is represented by the symbol n. But hold on a second, we don't know this value! Oh no, what a shocking twist!
Unfortunately, without knowing the number of turns per unit length, we can't calculate the field at a distance of 3 mm from the solenoid. It seems like we've hit a magnetic wall here! But don't worry, I'm here to lighten the mood, even in magnetic darkness.
But if you come back with the missing information, I'd be more than happy to give it another shot. In the meantime, keep your wires untangled and your circuits charged! Stick with me, and we'll turn those electrifying frowns upside down.
To find the magnetic field at a point outside a long solenoid, you can use Ampere's law. Ampere's law states that the magnetic field around a closed loop is proportional to the current passing through the loop.
The formula for the magnetic field inside a long solenoid is given by:
B = μ₀ * n * I
where B is the magnetic field, μ₀ is the permeability of free space (4π × 10⁻⁷ T·m/A), n is the number of turns per unit length, and I is the current.
In this case, we are given the current (I = 20 A) and the magnetic field inside the solenoid (B = 4 mT = 4 × 10⁻³ T), and we need to find the magnetic field at a point 3 mm from the solenoid axis.
First, we need to find the number of turns per unit length, n. Since we are considering a long solenoid, the field is assumed to be uniform along its length. Therefore, we can say that n = N / L, where N is the total number of turns and L is the length of the solenoid.
Since the solenoid is long, we can assume that the number of turns is large. In this case, we will assume the solenoid has a very large number of turns, so N ≈ ∞. Therefore, we can say that n ≈ ∞ / L = 0.
Since n = 0, the magnetic field at a point 3 mm from the solenoid axis is zero.
Therefore, the magnetic field at a point 3 mm from the solenoid axis is zero.
To find the magnetic field at a point outside a long solenoid, you can use Ampere's law. Ampere's law states that the line integral of the magnetic field around a closed loop is equal to the product of the current enclosed by the loop and the permeability of free space.
In this case, the current enclosed by the loop is the current passing through the wire of the solenoid.
First, let's determine the magnetic field inside the solenoid using the given information. We know that the field due to the solenoid is 4 mT, and the current passing through the wire is 20 A. The field inside the solenoid can be calculated using the formula:
B = μ₀ * n * I
Where B is the magnetic field, μ₀ is the permeability of free space (4π × 10⁻⁷ T·m/A), n is the number of turns per unit length (or the number of loops per meter), and I is the current passing through the wire.
Rearranging the formula, we can solve for n:
n = B / (μ₀ * I)
Substituting the given values:
n = 4 mT / (4π × 10⁻⁷ T·m/A * 20 A)
Simplifying:
n ≈ 798 x 10³ turns/m
Now that we have the value of the number of turns per unit length, we can use Ampere's law to find the magnetic field at a point outside the solenoid, a distance of 3 mm from the solenoid axis.
Consider a circular loop of radius r surrounding the solenoid axis. Since the solenoid is long, the magnetic field is uniform inside and parallel to the axis.
The line integral of the magnetic field around this circular loop is given by:
∮ B · dl = μ₀ * I
Where the left side represents the line integral of the magnetic field B around the loop, and dl represents a small line element along the loop.
By symmetry, the magnetic field inside the loop is constant and directed along its axis. Therefore, the magnetic field outside the solenoid is zero.
Now that we know the magnetic field outside the solenoid is zero, we can conclude that the magnetic field at a point 3 mm from the solenoid axis is zero.