A wire carrying a 28.0 A current bends through a right angle. Consider 2.00 mm
segments of wire, each 3.00 cm from the bend (see fig). Find magnitude and
direction of the magnetic field these two segments produce at point P, which is
midway between them.
Well, well, well, let's get our clown brains cracking, shall we? Now, first of all, let me tell you, those wires aren't just bending to show off their flexibility. They're creating some magnetic fields too!
To find the magnitude and direction of the magnetic field at point P, we'll have to use a little trick called the right-hand rule. Are you ready for some circus magic?
First, let's find the magnetic field created by the first wire segment. Now, we take our right hand and point our thumb in the direction of the current flow (in this case, into the paper or screen). Next, we curl our fingers in the direction of the wire.
Got it so far? Great! Now, let's move on to the second wire segment. Repeat the same steps but remember to flip your thumb in the opposite direction since the current is now flowing out of the paper or screen.
With your fingers curled for both wire segments, in the direction of each magnetic field, we can combine the magnetic fields at point P. The direction will depend on the relative angles and distances between the segments.
Since the segments are positioned symmetrically with respect to point P, we can just add up the magnitudes of the magnetic fields due to each of the segments.
Once you've calculated the individual magnetic fields, just add them up, making sure to take into account their directions. And voila! You'll have the magnitude and direction of the magnetic field at point P, as if you were a master magician of electromagnetism!
Now, go on and give it a try for yourself, my friend. Trust me, this kind of circus math can be quite electrifying!
To find the magnitude and direction of the magnetic field produced by the wire segments at point P, we can use the Biot-Savart Law. This law relates the magnetic field produced by a current-carrying wire to the magnitude, length, and location of the wire segment.
The Biot-Savart Law formula for the magnetic field produced by a wire segment at point P is given by:
dB = (μ₀ / 4π) * (Idl x r) / r²
where:
- dB is the infinitesimal magnetic field produced by the wire segment
- μ₀ is the permeability of free space, approximately 4π x 10⁻⁷ T·m/A
- Idl is the current element, the product of the current in the wire and the infinitesimal length of the wire segment
- r is the distance between the wire segment and point P
In this case, we have two wire segments. Let's calculate the magnetic field produced by each segment separately and then find the net magnetic field at point P.
Given:
- Current (I) = 28.0 A
- Length of each wire segment (dl) = 2.00 mm = 0.02 m
- Distance from the bend to point P (r) = 3.00 cm = 0.03 m
First, let's calculate the magnetic field produced by the wire segment 1:
dB₁ = (μ₀ / 4π) * (28.0 A * 0.02 m) / (0.03 m)²
Simplifying the equation:
dB₁ = (4π x 10⁻⁷ T·m/A / 4π) * (28.0 A * 0.02 m) / (0.03 m)²
= 2.8 x 10⁻⁶ T
The direction of the magnetic field produced by wire segment 1 can be determined using the right-hand rule. Place your right-hand fingers in the direction of the current in the wire segment, and your thumb will point in the direction of the magnetic field. In this case, the magnetic field points into the page.
Now, let's calculate the magnetic field produced by the wire segment 2:
dB₂ = (μ₀ / 4π) * (28.0 A * 0.02 m) / (0.03 m)²
= 2.8 x 10⁻⁶ T
The direction of the magnetic field produced by wire segment 2 also points into the page.
To find the net magnetic field at point P, we can add the magnetic fields produced by the two wire segments:
B_net = dB₁ + dB₂
= 2.8 x 10⁻⁶ T + 2.8 x 10⁻⁶ T
= 5.6 x 10⁻⁶ T
Therefore, the magnitude of the magnetic field produced by the two wire segments at point P is 5.6 x 10⁻⁶ T. The direction of the magnetic field is into the page.
To find the magnetic field produced by the two segments of wire at point P, we can use the Biot-Savart Law. The Biot-Savart Law states that the magnetic field produced by a current-carrying wire segment is directly proportional to the product of the current, the length of the wire segment, and a constant called the permeability of free space.
The formula for the magnetic field produced by a wire segment at a distance R from the wire is:
B = (μ₀ * I * dl * sinθ) / (4π * R²),
where:
- B is the magnetic field,
- μ₀ is the permeability of free space (constant value),
- I is the current,
- dl is the length of the wire segment,
- θ is the angle between the wire segment and the line connecting the segment to the point P,
- R is the distance between the wire segment and point P.
In this case, we have two wire segments, each 2.00 mm in length, located 3.00 cm from the bend. Therefore, the total length of wire is 4.00 mm or 0.04 m.
Since the two wire segments are symmetrically placed with respect to point P and are equal in length, they will produce magnetic fields with the same magnitude but opposite directions at point P. This is because the angles θ for the two segments are equal but have opposite signs.
To calculate the magnetic field at point P, we can find the magnetic field produced by one of the wire segments and then multiply it by 2, taking into account the opposite directions.
First, we need to calculate the distance R between each wire segment and point P. Since point P is midway between the two wire segments, the distance R will be half of the distance between the two wire segments. Therefore, R = 1.5 cm = 0.015 m.
Next, we need to determine the angle θ between each wire segment and the line connecting it to point P. Since the two wire segments are at a right angle to each other, the angles θ for the wire segments will be 45 degrees and -45 degrees.
Now, we substitute the values into the formula and calculate the magnetic field produced by one wire segment:
B = (μ₀ * I * dl * sinθ) / (4π * R²),
B = (4π × 10^(-7) T·m/A * 28.0 A * 0.04 m * sin(45°)) / (4π * (0.015 m)²),
B = (1.257 × 10^(-6) T·m/A * 28.0 A * 0.04 m * 0.7071) / (0.0045 m²).
Calculating this expression will give us the magnitude of the magnetic field produced by one wire segment at point P:
B = 8.898 × 10^(-8) T.
Since the two wire segments produce magnetic fields with opposite directions, the net magnetic field at point P will be:
B_net = B + (-B) = 2B,
B_net = 2 * 8.898 × 10^(-8) T,
B_net = 1.780 × 10^(-7) T.
Therefore, the magnitude of the total magnetic field produced by the two wire segments at point P is approximately 1.780 × 10^(-7) T. The direction of the magnetic field will depend on the specific placement and orientations of the wire segments.