15.(16分）如图所示，在xOy坐标平面内，半径为 R 的圆形匀强磁场区域与x 轴相切于原点O，与PM相切于A点，PQNM 为第一象限内边长为2R、下边在ェ轴上的正方形，其内部有沿y轴正方向的匀强电场。现有大量质量为 m、电荷量为q的正离子，从O点以相同的速率v。沿纸面均匀向各个方向射出，进入磁场的离子从磁场边界出射的点分布在三分之一的圆周上，离子到达MN边界即被吸收，不计离子受到的重力及离子间的相互作用。(1）求匀强磁场的磁感应强度大小B;
(2）若仅改变磁感应强度的大小，使得其中沿y轴正方向人射的离子能经过A点打到N 点，求电场强度的大小E;
(3）在（2）的基础上，仅将电场强度的大小调整为一E，求MN 区域接收到的离子数占发射出的总离子数的比例7。
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15. (16 分) 如图所示, 在 xOy 坐标平面内, 半径为 R 的圆形匀强磁场区域与 x 轴相切于原点 O, 与 PM 相切于 A 点, PQNM 为第一象限内边长为 2R、 下边在 x 轴上的正方形, 其内部有沿 y 轴正方向的匀强电场。现有大量质量为 m、电荷量为 q 的正离子, 从 O 点以相同的速率 v₀ 沿纸面内均匀向各个方向射出, 进入磁场的离子从磁场边界射出的点分布在三分之一 的圆周上, 离子到达 MN 边界即被吸收, 不计离子受到的重力及离子间的相互作用。
(1) 求匀强磁场的磁感应强度大小 B;
(2) 若仅改变磁感应强度的大小, 使得其中沿 y 轴正方向入射的离子能经过 A 点打到 N 点, 求电场强度的大小 E;
(3) 在 (2) 的基础上, 仅将电场强度的大小调整为 1/2 E, 求 MN 区域接收到的离子数占发射出 的总离子数的比例 η。
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**Diagram Description:**
* **Type:** Coordinate plane showing regions of magnetic and electric fields and particle trajectories.
* **Coordinate Axes:** x-axis (horizontal, positive to the right), y-axis (vertical, positive upwards). Origin is labeled O.
* **Regions:**
* Circular region in the first quadrant, labeled as a uniform magnetic field. It is tangent to the x-axis at O. It appears centered on the y-axis. Based on tangency at O (0,0) and radius R, the center is at (0, R). The boundary is the circle x² + (y-R)² = R².
* Square region PQNM in the first quadrant, labeled as a uniform electric field. Its bottom side PQ is on the x-axis. P is to the right of O. The side length is 2R.
* **Points:**
* O: Origin (0,0). Point where magnetic field is tangent to x-axis. Point of ion emission.
* A: Point on the boundary of the circular magnetic field region and on the line segment PM (left side of the square). The circle is tangent to the line PM at A. Given circle center (0,R) and radius R, and PM being a vertical line x=constant, tangency implies the x-coordinate of A and PM is R. A is on the circle, so R² + (y_A - R)² = R², which gives y_A = R. Thus, A is at (R, R).
* P: Bottom-left corner of the square, on the x-axis. Since PM is at x=R, P is at (R, 0).
* Q: Bottom-right corner of the square, on the x-axis. Side length is 2R, so Q is at (R+2R, 0) = (3R, 0).
* M: Top-left corner of the square. Above P, side length 2R. M is at (R, 2R).
* N: Top-right corner of the square. Above Q, side length 2R. N is at (3R, 2R).
* **Lines/Segments:**
* PQ: Segment on x-axis from (R,0) to (3R,0).
* PM: Vertical segment from (R,0) to (R,2R). Left side of the square.
* QN: Vertical segment from (3R,0) to (3R,2R). Right side of the square.
* MN: Horizontal segment from (R,2R) to (3R,2R). Top side of the square. This is the absorption boundary.
* **Magnetic Field:** Indicated by the circular region and possibly dots inside (suggesting uniformity). Direction is not explicitly shown by arrows but is implied to be perpendicular to the plane for circular motion. Let's assume it's out of the page (B > 0).
* **Electric Field:** Indicated by vertical upward arrows inside the square PQNM. Uniform and along the positive y-axis (E > 0).
* **Particles:** Positive ions (charge q, mass m) emitted from O with speed v₀ in all directions.
* **Annotations:** R (radius/side length), v₀ (initial speed), m (mass), q (charge), B (magnetic field magnitude), E (electric field magnitude), η (ratio).
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**Extracted Content:**
15. (16 分) 如图所示, 在 xOy 坐标平面内, 半径为 R 的圆形匀强磁场区域与 x 轴相切于原点 O, 与 PM 相切于 A 点, PQNM 为第一象限内边长为 2R、 下边在 x 轴上的正方形, 其内部有沿 y 轴正方向的匀强电场。现有大量质量为 m、电荷量为 q 的正离子, 从 O 点以相同的速率 v₀ 沿纸面内均匀向各个方向射出, 进入磁场的离子从磁场边界射出的点分布在三分之一 的圆周上, 离子到达 MN 边界即被吸收, 不计离子受到的重力及离子间的相互作用。
(1) 求匀强磁场的磁感应强度大小 B;
(2) 若仅改变磁感应强度的大小, 使得其中沿 y 轴正方向入射的离子能经过 A 点打到 N 点, 求电场强度的大小 E;
(3) 在 (2) 的基础上, 仅将电场强度的大小调整为 $\frac{1}{2}E$, 求 MN 区域接收到的离子数占发射出 的总离子数的比例 η。
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